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