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Fourier integral operators and the index of symplectomorphisms on manifolds with boundary
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Window-dependent bases for efficient representations of the Stockwell transform
Full text linkSince its appearing in 1996, the Stockwell transform (S-transform) has been used as a tool in medical imaging, geophysics and signal processing in general. In this paper, we prove that the system of functions (so-called DOST basis) is indeed an orthonormal basis of L2 pr0, 1sq, which is a time-frequency localized basis, in the sense of Donoh-Stark Theorem [11]. Our approach provides a unified setting in which to study the Stockwell transform and its orthogonal decomposition. Finally, we introduce a fast – O pN logNq – algorithm to compute the Stockwell coefficients for general windows. This algorithm includes the one proposed in [33] as a special case
On the application of the Stockwell transform to GPR data analysis
Full text linkFor most archaeological prospections with GPR, target detection is the most important aspect of the surveys. The main efforts of GPR data processing are committed to increase the signal to noise ratio in radargrams. The usual processing aims to: filter the radar section from clutters and attenuate the in-band noises; enhance coherent reflections that are likely due to the target response; transform the acquired time section into a depth section; build a 3D image of the reflection targets in the subsoil by correlating reflections from adjacent radargrams. In this paper we show some application of the Stockwell transform, both to synthetic and field data, to enhance the signal to noise ratio in radargrams
A Class of Fourier Integral Operators on Manifolds with Boundary
No full textWe study a class of Fourier integral operators on compact manifolds with boundary, associated with a natural class of symplectomorphisms, namely, those which preserve the boundary. A calculus of Boutet de Monvel's type can be defined for such Fourier integral operators, and appropriate continuity properties established. One of the key features of this calculus is that the local representations of these operators are given by operator-valued symbols acting on Schwartz functions or temperate distributions. Here we focus on properties of the corresponding local phase functions, which allow to prove this result in a rather straightforward wa
Energy transfer between modes in a nonlinear beam equation
Full text linkWe consider the nonlinear nonlocal beam evolution equation introduced by Woinowsky-Krieger We study the existence and behavior of periodic solutions: these are called nonlinear modes. Some solutions only have two active modes and we investigate whether there is an energy transfer between them. The answer depends on the geometry of the energy function which, in turn, depends on the amount of compression compared to the spatial frequencies of the involved modes. Our results are complemented with numerical experiments; overall, they give a complete picture of the instabilities that may occur in the beam. We expect these results to hold also in more complicated dynamical systems. On considà ̈re l'équation d'évolution de la poutre non linéaire et non locale introduite par Woinowsky-Krieger On étudie l'existence et le comportement des solutions périodiques: on les appelle modes non linéaires. Certaines solutions ont seulement deux modes actifs et on étudie le possible transfert d'énergie entre eux. La réponse dépend de la géométrie de la fonctionnelle d'énergie qui, à son tour, dépend de la quantité de compression et des fréquences spatiales des modes actifs. Les résultats sont complétés par des expérimentations numériques ; ils donnent une description d'ensemble assez complà ̈te des instabilités qui peuvent apparaître dans la poutre. On s'attend à ce que ces résultats soient valables aussi pour des systà ̈mes dynamiques plus compliqués
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