M.Sc Student | Meir Alon |
---|---|
Subject | Gabor-Type Expansion with Non-Exponential Kernels |
Department | Department of Electrical Engineering | Supervisors | Dr. Meir Zibulski |
Professor Emeritus Yehoshua Zeevi |
In
his original optimal scheme of signal representation, Gabor implemented a
Gaussian window and critical sampling density of the combined time-frequency
(TF) space, and a complex-exponential, harmonic, kernel. This expansion was
later investigated using different window functions, and/or sampling densities,
and/or kernel functions. In the present work we investigate the application of
two specific, non-standard, families of kernel functions.
The
window function proposed by Gabor is the best in terms of the joint entropy,
i.e. the compactness of the representation in the combined space. However,
using this window function, or any other "well-behaved" localized
function, with critical sampling of the combined Gabor space, entails a problem
of stability, i.e., the set of Gabor elementary functions do not constitute a
frame.
The
purpose of this work is to examine advantages afforded by two specific families
of non-exponential kernel functions. In particular, a "well
localized" window function is used with these kernels in order to examine
whether such a combination can overcome the problem of representation
stability, inherent in the case of critical sampling.
We
show that there is a solution to the problem of stability in the case of
critical sampling by using unconventional kernel. In the case of the first
kernel proposed in this study, we extract one of the unique properties of the
solution, and generalize it with the goal to achieve stability. The set of
Gabor elementary function does not constitute, in this case, a frame over the
interval of its periodicity, but it does constitute a frame over subinterval of
the periodicity. In the case of the second proposed kernel, we attempt to
achieve stability by using variable resolution over the TF combined space,
having low resolution for high frequencies and high resolution for low
frequencies.
Using
the first type of non-exponential kernel, we propose new ways of how to
construct the best representation set for a given implementation. This is done
by improving the frame properties and the TF localization of the representation
set. It is shown that the localization in the frequency domain can be improved
without effecting the time domain localization, but it can not be done in such
a way that the frequency representation of the proposed elementary functions
will be square integrable. Lastly, we also address the issue of controlling the
redundancy in the L2(R) spanning set {gmn(x)}
by selection of the appropriate kernel function.