83 research outputs found
Linear perturbations of the Wigner distribution and the Cohen class
The Wigner distribution is a milestone of Time-frequency Analysis. In order to cope with its drawbacks while preserving the desirable features that made it so popular, several kinds of modifications have been proposed. This contribution fits into this perspective. We introduce a family of phase-space representations of Wigner type associated with invertible matrices and explore their general properties. As a main result, we provide a characterization for the Cohen's class [L. Cohen, Generalized phase-space distribution functions, J. Math. Phys. 7 (1996) 781-786; Time-frequency Analysis (Prentice Hall, New Jersey, 1995)]. This feature suggests to interpret this family of representations as linear perturbations of the Wigner distribution. We show which of its properties survive under linear perturbations and which ones are truly distinctive of its central role
Linear perturbations of the Wigner transform and the Weyl quantization
We study a class of quadratic time-frequency representations that, roughly
speaking, are obtained by linear perturbations of the Wigner transform. They
satisfy Moyal's formula by default and share many other properties with the
Wigner transform, but in general they do not belong to Cohen's class. We
provide a characterization of the intersection of the two classes. To any such
time-frequency representation, we associate a pseudodifferential calculus. We
investigate the related quantization procedure, study the properties of the
pseudodifferential operators, and compare the formalism with that of the Weyl
calculus.Comment: 38 pages. Contributed chapter for volume on the occasion of Luigi
Rodino's 70th birthda
Generalized Born--Jordan Distributions and Applications
The quadratic nature of the Wigner distribution causes the appearance of
unwanted interferences. This is the reason why engineers, mathematicians and
physicists look for related time-frequency distributions, many of them are
members of the Cohen class. Among them, the Born-Jordan distribution has
recently attracted the attention of many authors, since the so-called ghost
frequencies are damped quite well, and the noise is in general reduced. The
very insight relies on the kernel of such a distribution, which contains the
sinus cardinalis (sinc), which can be viewed as the Fourier transform, of the
first B-Spline. Replacing the function B-Spline with the spline or order n, on
the Fourier side we obtain the n-th power of sinc, whose decay at infinity
increases with n. We introduce the corresponding Cohen's Kernel and study the
properties of the related time-frequency distribution which we call generalized
Born--Jordan distribution
Mixed-state localization operators: Cohen's class and trace class operators
We study mixed-state localization operators from the perspective of Werner's
operator convolutions which allows us to extend known results from the rank-one
case to trace class operators. The idea of localizing a signal to a domain in
phase space is approached from various directions such as bounds on the
spreading function, probability densities associated to mixed-state
localization operators, positive operator valued measures, positive
correspondence rules and variants of Tauberian theorems for operator
translates. Our results include a rigorous treatment of multiwindow-STFT
filters and a characterization of mixed-state localization operators as
positive correspondence rules. Furthermore, we provide a description of the
Cohen class in terms of Werner's convolution of operators and deduce
consequences on positive Cohen class distributions, an uncertainty principle,
uniqueness and phase retrieval for general elements of Cohen's class.Comment: We call generalized localization operators now mixed-state
localization operators. In addition to a change of title and other parts
involving generalized localization operators. We did a major revision of the
manuscript incorporating suggestions by reviewer
Decoherence and entropy of primordial fluctuations. I: Formalism and interpretation
We propose an operational definition of the entropy of cosmological
perturbations based on a truncation of the hierarchy of Green functions. The
value of the entropy is unambiguous despite gauge invariance and the
renormalization procedure. At the first level of truncation, the reduced
density matrices are Gaussian and the entropy is the only intrinsic quantity.
In this case, the quantum-to-classical transition concerns the entanglement of
modes of opposite wave-vectors, and the threshold of classicality is that of
separability. The relations to other criteria of classicality are established.
We explain why, during inflation, most of these criteria are not intrinsic. We
complete our analysis by showing that all reduced density matrices can be
written as statistical mixtures of minimal states, the squeezed properties of
which are less constrained as the entropy increases. Pointer states therefore
appear not to be relevant to the discussion. The entropy is calculated for
various models in paper II.Comment: 23 page
Wigner analysis of operators. Part I: pseudodifferential operators and wave fronts
We perform Wigner analysis of linear operators. Namely, the standard
time-frequency representation \emph{Short-time Fourier Transform} (STFT) is
replaced by the -\emph{Wigner distribution} defined by
, where
is a symplectic matrix and is an associate
metaplectic operator. Basic examples are given by the so-called -Wigner
distributions. Such representations provide a new characterization for
modulation spaces when . Furthermore, they can be efficiently
employed in the study of the off-diagonal decay for pseudodifferential
operators with symbols in the Sj\"ostrand class (in particular, in the
H\"{o}rmander class ). The novelty relies on defining time-frequency
representations via metaplectic operators, developing a conceptual framework
and paving the way for a new understanding of quantization procedures. We
deduce micro-local properties for pseudodifferential operators in terms of the
Wigner wave front set. Finally, we compare the Wigner with the global
H\"{o}rmander wave front set and identify the possible presence of a ghost
region in the Wigner wave front. \par In the second part of the paper
applications to Fourier integral operators and Schr\"{o}dinger equations will
be given.Comment: Improved version. 43 page
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