Kamilla Haahr Nielsen: In harmonic analysis, frames offer a way of representing complex signals using much simpler building blocks. Compared to orthonormal bases (ONB), frames introduce redundancy which provides robustness, e.g., when transmitting signals.
In applied mathematics and engineering, frames offer a way of representing complex signals using a system of much simpler building blocks. Compared to orthonormal bases, frames introduce redundancy which provides robustness, e.g., when transmitting signals. This thesis considers concrete systems of functions, so-called Gabor systems, that are constructed by taking translations and modulations of a fixed window function. It is a well-known in harmonic analysis fact that it is not possible to find Gabor orthonormal bases generated by "nice" window functions and that the limitations disappears when considering Gabor frames.
In applications, it is essential for the decomposition and reconstruction process to have a so-called pair of dual frames as dual frames are used to generate stable expansions of signals with easily computable coefficients. The thesis is concerned with dual Gabor windows with perfect time-localization, i.e., dual Gabor frames with compactly supported window functions. The thesis develops a number of methods for constructing compactly supported dual windows with desirable properties such as smoothness, symmetry, boundedness and small support. The main advantage of the construction methods is that they allow for the dual window to inherit the same level of smoothness as the original window function. The latter property is crucial for frequency-localization of the windows and important in many applications.
PhD project title: Constructing dual windows for Gabor frames
Effective start/end date 01/08/2016 → 31/07/2019
Supervisors: Ole Christensen, Jakob Lemvig, Section for Mathematics