457 research outputs found
Strong and weak semiclassical limits for some rough Hamiltonians
We present several results concerning the semiclassical limit of the time
dependent Schr\"odinger equation with potentials whose regularity doesn't
guarantee the uniqueness of the underlying classical flow. Different topologies
for the limit are considered and the situation where two bicharateristics can
be obtained out of the same initial point is emphasized
Remarks on the notion of time-frequency localization
The idea behind a time-frequency representation is often connected with that of some form of localization in the plane . This point o f
view is faced with a number of limitations, which all express in some way uncertainty relations between time and frequency . In the
case of << chirps », there exist however adapted representations which can localize very accurately on specific curves of the plane ,
and whose construction can be merely justified on the basis of geometric arguments . As a corollary, such joint descriptions, in both
time and frequency, allow for an interpretaion of the corresponding localization curves in terms of « instantaneous frequencies» .L'idée de représentation temps-fréquence d'un signal est en général associée à celle d'une forme de localisation dans le plan. Ce point de vue se heurte à un certain nombre de limites qui sont autant de façons d'exprimer des relations d'incertitude entre le temps et la fréquence. Dans le cas de signaux « chirps », des représentations adaptées peuvent néanmoins se localiser de façon très précise sur des courbes spécifiques du plan, la construction de telles représentations pouvant se faire sur la base d'arguments essentiellement géométriques. Décrire un signal conjointement en temps et en fréquence permet en corollaire d'interpréter les courbes sur lesquelles la représentation se localise en termes de « fréquences instantanées »
Time-frequency representations for non-stationary signals
Spectral analysis of non-stationary signais calls for specific tools which permit one to describe a rime evolution of frequency
characteristics . Such tools, referred to as time frequency representations, can be defined in an objective way when imposing
a priori constraints . Within a stochastic and non parametric framework, two main approaches are offered, which either emphasize
a doubly orthogonal decomposition, or preserve the usual concept of frequency. After having established the corresponding
definitions and emphasized the importance of the Wigner-Ville transform, estimation problems are addressed and a discussion is
provided for supporting the usefulness of time frequency representations in processing operations which go beyond a mere
description.Etablissement de la transformation de Wigner-Ville, étude des possibilités d'estimation et discussion sur une représentation temps-fréquence pouvant être utilisée pour des opérations de traitement dépassant la seule descriptio
Recent Advances in Theory and Methods for Nonstationary Signal Analysis
Cataloged from PDF version of article.All physical processes are nonstationary. When analyzing
time series, it should be remembered that nature can
be amazingly complex and that many of the theoretical
constructs used in stochastic process theory, for example,
linearity, ergodicity, normality, and particularly stationarity,
are mathematical fairy tales. There are no stationary time
series in the strict mathematical sense; at the very least, everything
has a beginning and an end. Thus, while it is necessary
to know the theory of stationary processes, one should not
adhere to it dogmatically when analyzing data from physical
sources, particularly when the observations span an extended
period. Nonstationary signals are appropriate models for
signals arising in several fields of applications including
communications, speech and audio, mechanics, geophysics,
climatology, solar and space physics, optics, and biomedical
engineering. Nonstationary models account for possible time
variations of statistical functions and/or spectral characteristics
of signals. Thus, they provide analysis tools more general
than the classical Fourier transform for finite-energy signals
or the power spectrum for finite-power stationary signals.
Nonstationarity, being a “nonproperty” has been analyzed
from several different points of view. Several approaches
that generalize the traditional concepts of Fourier analysis
have been considered, including time-frequency, time-scale,
and wavelet analysis, and fractional Fourier and linear
canonical transforms
Recent Advances in Theory and Methods for Nonstationary Signal Analysis
Cataloged from PDF version of article.All physical processes are nonstationary. When analyzing
time series, it should be remembered that nature can
be amazingly complex and that many of the theoretical
constructs used in stochastic process theory, for example,
linearity, ergodicity, normality, and particularly stationarity,
are mathematical fairy tales. There are no stationary time
series in the strict mathematical sense; at the very least, everything
has a beginning and an end. Thus, while it is necessary
to know the theory of stationary processes, one should not
adhere to it dogmatically when analyzing data from physical
sources, particularly when the observations span an extended
period. Nonstationary signals are appropriate models for
signals arising in several fields of applications including
communications, speech and audio, mechanics, geophysics,
climatology, solar and space physics, optics, and biomedical
engineering. Nonstationary models account for possible time
variations of statistical functions and/or spectral characteristics
of signals. Thus, they provide analysis tools more general
than the classical Fourier transform for finite-energy signals
or the power spectrum for finite-power stationary signals.
Nonstationarity, being a “nonproperty” has been analyzed
from several different points of view. Several approaches
that generalize the traditional concepts of Fourier analysis
have been considered, including time-frequency, time-scale,
and wavelet analysis, and fractional Fourier and linear
canonical transforms
An Entropy Based Method for Local Time-Adaptation of the Spectrogram
We propose a method for automatic local time-adaptation of the spectrogram of
audio signals: it is based on the decomposition of a signal within a Gabor
multi-frame through the STFT operator. The sparsity of the analysis in every
individual frame of the multi-frame is evaluated through the R\'enyi entropy
measures: the best local resolution is determined minimizing the entropy
values. The overall spectrogram of the signal we obtain thus provides local
optimal resolution adaptively evolving over time. We give examples of the
performance of our algorithm with an instrumental sound and a synthetic one,
showing the improvement in spectrogram displaying obtained with an automatic
adaptation of the resolution. The analysis operator is invertible, thus leading
to a perfect reconstruction of the original signal through the analysis
coefficients
Localization of Multi-Dimensional Wigner Distributions
A well known result of P. Flandrin states that a Gaussian uniquely maximizes
the integral of the Wigner distribution over every centered disc in the phase
plane. While there is no difficulty in generalizing this result to
higher-dimensional poly-discs, the generalization to balls is less obvious. In
this note we provide such a generalization.Comment: Minor corrections, to appear in the Journal of Mathematical Physic
In situ non-invasive Raman spectroscopic characterisation of succinic acid polymorphism during segmented flow crystallisation
The kinetically regulated automated input crystalliser for Raman spectroscopy (KRAIC-R) combines highly controlled crystallisation environments, via tri segmented flow, with non-invasive confocal Raman spectroscopy. Taking advantage of the highly reproducible crystallisation environment within a segmented flow crystalliser and the non-invasive nature of confocal spectroscopy, we are able to shine light on the nucleation and growth of Raman active polymorphic materials without inducing unrepresentative crystallisation events through our analysis technique. Using the KRAIC-R we have probed the nucleation and subsequent growth of succinic acid. Succinic acid typically crystallises as β-SA from solution-based crystallisation although some examples of a small proportion of α-SA have been reported in the β-SA product. Here we show that α-SA and β-SA nucleate concomitantly but undergo Ostwald ripening to a predominantly β-SA product
Acoustic characterization of Hofstadter butterfly with resonant scatterers
We are interested in the experimental characterization of the Hofstadter
butterfly by means of acoustical waves. The transmission of an acoustic pulse
through an array of 60 variable and resonant scatterers periodically distribued
along a waveguide is studied. An arbitrary scattering arrangement is realized
by using the variable length of each resonator cavity. For a periodic
modulation, the structures of forbidden bands of the transmission reproduce the
Hofstadter butterfly. We compare experimental, analytical, and computational
realizations of the Hofstadter butterfly and we show the influence of the
resonances of the scatterers on the structure of the butterfly
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