3,923 research outputs found
Designing Gabor windows using convex optimization
Redundant Gabor frames admit an infinite number of dual frames, yet only the
canonical dual Gabor system, constructed from the minimal l2-norm dual window,
is widely used. This window function however, might lack desirable properties,
e.g. good time-frequency concentration, small support or smoothness. We employ
convex optimization methods to design dual windows satisfying the Wexler-Raz
equations and optimizing various constraints. Numerical experiments suggest
that alternate dual windows with considerably improved features can be found
Gabor representations of evolution operators
We perform a time-frequency analysis of Fourier multipliers and, more
generally, pseudodifferential operators with symbols of Gevrey, analytic and
ultra-analytic regularity. As an application we show that Gabor frames, which
provide optimally sparse decompositions for Schroedinger-type propagators,
reveal to be an even more efficient tool for representing solutions to a wide
class of evolution operators with constant coefficients, including weakly
hyperbolic and parabolic-type operators. Besides the class of operators, the
main novelty of the paper is the proof of super-exponential (as opposite to
super-polynomial) off-diagonal decay for the Gabor matrix representation.Comment: 26 page
An optimally concentrated Gabor transform for localized time-frequency components
Gabor analysis is one of the most common instances of time-frequency signal
analysis. Choosing a suitable window for the Gabor transform of a signal is
often a challenge for practical applications, in particular in audio signal
processing. Many time-frequency (TF) patterns of different shapes may be
present in a signal and they can not all be sparsely represented in the same
spectrogram. We propose several algorithms, which provide optimal windows for a
user-selected TF pattern with respect to different concentration criteria. We
base our optimization algorithm on -norms as measure of TF spreading. For
a given number of sampling points in the TF plane we also propose optimal
lattices to be used with the obtained windows. We illustrate the potentiality
of the method on selected numerical examples
A few remarks on time-frequency analysis of Gevrey, analytic and ultra-analytic functions
We give a brief survey of recent results concerning almost diagonalization of
pseudodifferential operators via Gabor frames. Moreover, we show new
connections between symbols with Gevrey, analytic or ultra-analityc regularity
and time-frequency analysis of the corresponding pseudodifferential operators.Comment: 16 pages. arXiv admin note: text overlap with arXiv:1209.094
A survey of uncertainty principles and some signal processing applications
The goal of this paper is to review the main trends in the domain of
uncertainty principles and localization, emphasize their mutual connections and
investigate practical consequences. The discussion is strongly oriented
towards, and motivated by signal processing problems, from which significant
advances have been made recently. Relations with sparse approximation and
coding problems are emphasized
Density and duality theorems for regular Gabor frames
We investigate Gabor frames on locally compact abelian groups with
time-frequency shifts along non-separable, closed subgroups of the phase space.
Density theorems in Gabor analysis state necessary conditions for a Gabor
system to be a frame or a Riesz basis, formulated only in terms of the index
subgroup. In the classical results the subgroup is assumed to be discrete. We
prove density theorems for general closed subgroups of the phase space, where
the necessary conditions are given in terms of the "size" of the subgroup. From
these density results we are able to extend the classical Wexler-Raz
biorthogonal relations and the duality principle in Gabor analysis to Gabor
systems with time-frequency shifts along non-separable, closed subgroups of the
phase space. Even in the euclidean setting, our results are new
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