42 research outputs found
Multi-Window Weaving Frames
In this work we deal with the recently introduced concept of weaving frames.
We extend the concept to include multi-window frames and present the first
sufficient criteria for a family of multi-window Gabor frames to be woven. We
give a Hilbert space norm criterion and a pointwise criterion in phase space.
The key ingredient are localization operators in phase space and we give
examples of woven multi-window Gabor frames consisting of Hermite functions.Comment: 9 pages, conference paper: SampTA 201
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
Gabor frames and deep scattering networks in audio processing
This paper introduces Gabor scattering, a feature extractor based on Gabor
frames and Mallat's scattering transform. By using a simple signal model for
audio signals specific properties of Gabor scattering are studied. It is shown
that for each layer, specific invariances to certain signal characteristics
occur. Furthermore, deformation stability of the coefficient vector generated
by the feature extractor is derived by using a decoupling technique which
exploits the contractivity of general scattering networks. Deformations are
introduced as changes in spectral shape and frequency modulation. The
theoretical results are illustrated by numerical examples and experiments.
Numerical evidence is given by evaluation on a synthetic and a "real" data set,
that the invariances encoded by the Gabor scattering transform lead to higher
performance in comparison with just using Gabor transform, especially when few
training samples are available.Comment: 26 pages, 8 figures, 4 tables. Repository for reproducibility:
https://gitlab.com/hararticles/gs-gt . Keywords: machine learning; scattering
transform; Gabor transform; deep learning; time-frequency analysis; CNN.
Accepted and published after peer revisio
Compactness Criteria in Function Spaces
The classical criterion for compactness in Banach spaces of functions can be
reformulated into a simple tightness condition in the time-frequency domain.
This description preserves more explicitly the symmetry between time and
frequency than the classical conditions. The result is first stated and proved
for L^2(R^d), and then generalized to coorbit spaces. As special cases, we
obtain new characterizations of compactness in Besov-Triebel-Lizorkin spaces,
modulation spaces and Bargmann-Fock spaces
Representation of operators by sampling in the time-frequency domain
International audienceGabor multipliers are well-suited for the approximation of certain time-variant systems. However, this class of systems is rather restricted. To overcome this restriction, multiple Gabor multipliers allowing for more than one synthesis windows are introduced. The influence of the choice of the various parameters involved on approximation quality is studied for both classical and multiple Gabor multipliers
Spreading function representation of operators and Gabor multiplier approximation
Modification of signals in the time-frequency domain are used in many applications. However, the modification is often restricted to be purely multiplicative. In this paper, it is shown that, in the continuous case, a quite general class of operators can be represented by a twisted convolution in the short-time Fourier transform domain. The discrete case of Gabor transforms turns out to be more intricate. A similar representation will however be derived by means of a special form for the operator's spreading function (twisted spline type function). The connection between STFT- and Gabor-multipliers, their spreading function and the twisted convolution representation will be investigated. A precise characterization of the best approximation and its existence is given for both cases. Finally, the concept of Gabor multipliers is generalized to better approximate ''overspread'' operators