2,728 research outputs found
Gabor frames and asymptotic behavior of Schwartz distributions
We obtain characterizations of asymptotic properties of Schwartz distribution by using Gabor frames. Our characterizations are indeed Tauberian theorems for shift asymptotics (S-asymptotics) in terms of short-time Fourier transforms with respect to windows generating Gabor frames. For it, we show that the Gabor coefficient operator provides (topological) isomorphisms of the spaces of tempered distributions and distributions of exponential type onto their images
A Phase Vocoder based on Nonstationary Gabor Frames
We propose a new algorithm for time stretching music signals based on the
theory of nonstationary Gabor frames (NSGFs). The algorithm extends the
techniques of the classical phase vocoder (PV) by incorporating adaptive
time-frequency (TF) representations and adaptive phase locking. The adaptive TF
representations imply good time resolution for the onsets of attack transients
and good frequency resolution for the sinusoidal components. We estimate the
phase values only at peak channels and the remaining phases are then locked to
the values of the peaks in an adaptive manner. During attack transients we keep
the stretch factor equal to one and we propose a new strategy for determining
which channels are relevant for reinitializing the corresponding phase values.
In contrast to previously published algorithms we use a non-uniform NSGF to
obtain a low redundancy of the corresponding TF representation. We show that
with just three times as many TF coefficients as signal samples, artifacts such
as phasiness and transient smearing can be greatly reduced compared to the
classical PV. The proposed algorithm is tested on both synthetic and real world
signals and compared with state of the art algorithms in a reproducible manner.Comment: 10 pages, 6 figure
Superposition frames for adaptive time-frequency analysis and fast reconstruction
In this article we introduce a broad family of adaptive, linear
time-frequency representations termed superposition frames, and show that they
admit desirable fast overlap-add reconstruction properties akin to standard
short-time Fourier techniques. This approach stands in contrast to many
adaptive time-frequency representations in the extant literature, which, while
more flexible than standard fixed-resolution approaches, typically fail to
provide efficient reconstruction and often lack the regular structure necessary
for precise frame-theoretic analysis. Our main technical contributions come
through the development of properties which ensure that this construction
provides for a numerically stable, invertible signal representation. Our
primary algorithmic contributions come via the introduction and discussion of
specific signal adaptation criteria in deterministic and stochastic settings,
based respectively on time-frequency concentration and nonstationarity
detection. We conclude with a short speech enhancement example that serves to
highlight potential applications of our approach.Comment: 16 pages, 6 figures; revised versio
Gabor analysis over finite Abelian groups
The topic of this paper are (multi-window) Gabor frames for signals over
finite Abelian groups, generated by an arbitrary lattice within the finite
time-frequency plane. Our generic approach covers simultaneously
multi-dimensional signals as well as non-separable lattices. The main results
reduce to well-known fundamental facts about Gabor expansions of finite signals
for the case of product lattices, as they have been given by Qiu, Wexler-Raz or
Tolimieri-Orr, Bastiaans and Van-Leest, among others. In our presentation a
central role is given to spreading function of linear operators between
finite-dimensional Hilbert spaces. Another relevant tool is a symplectic
version of Poisson's summation formula over the finite time-frequency plane. It
provides the Fundamental Identity of Gabor Analysis.In addition we highlight
projective representations of the time-frequency plane and its subgroups and
explain the natural connection to twisted group algebras. In the
finite-dimensional setting these twisted group algebras are just matrix
algebras and their structure provides the algebraic framework for the study of
the deeper properties of finite-dimensional Gabor frames.Comment: Revised version: two new sections added, many typos fixe
Frame Constants of Gabor Frames near the Critical Density
We consider Gabor frames generated by a Gaussian function and describe the
behavior of the frame constants as the density of the lattice approaches the
critical value
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