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Benjamin Ricaud

Publications and source records attributed to Benjamin Ricaud.

26 records · Page 2Linked to original sources
arXiv2013

Vertex-Frequency Analysis on Graphs

One of the key challenges in the area of signal processing on graphs is to design dictionaries and transform methods to identify and exploit structure in signals on weighted graphs. To do so, we need to account for the intrinsic geometric structure of the underlying graph data domain. In this paper, we generalize one of the most important signal processing tools - windowed Fourier analysis - to the graph setting. Our approach is to first define generalized convolution, translation, and modulation operators for signals on graphs, and explore related properties such as the localization of translated and modulated graph kernels. We then use these operators to define a windowed graph Fourier transform, enabling vertex-frequency analysis. When we apply this transform to a signal with frequency components that vary along a path graph, the resulting spectrogram matches our intuition from classical discrete-time signal processing. Yet, our construction is fully generalized and can be applied to analyze signals on any undirected, connected, weighted graph.

math.FA↗
arXiv2012

Refined support and entropic uncertainty inequalities

Generalized versions of the entropic (Hirschman-Beckner) and support (Elad-Bruckstein) uncertainty principle are presented for frames representations. Moreover, a sharpened version of the support inequality has been obtained by introducing a generalization of the coherence. In the finite dimensional case and under certain conditions, minimizers of this inequalities are given as constant functions on their support. In addition, $\ell^p$-norms inequalities are introduced as byproducts of the entropic inequalities.

cs.IT↗
arXiv2010

Properties of periodic solutions near their oscillation threshold for a class of hyperbolic partial differential equations with localized nonlinearity

The periodic solutions of a type of nonlinear hyperbolic partial differential equations with a localized nonlinearity are investigated. For instance, these equations are known to describe several acoustical systems with fluid-structure interaction. It also encompasses particular types of delay differential equations. These systems undergo a bifurcation with the appearance of a small amplitude periodic regime. Assuming a certain regularity of the oscillating solution, several of its properties around the bifurcation are given: bifurcation point, dependence of both the amplitude and period with respect to the bifurcation parameter, and law of decrease of the Fourier series components. All the properties of the standard Hopf bifurcation in the non-hyperbolic case are retrieved. In addition, this study is based on a Fourier domain analysis and the harmonic balance method has been extended to the class of infinite dimensional problems hereby considered. Estimates on the errors made if the Fourier series is truncated are provided.

math.DS↗
arXiv2010

Analysis and separation of time-frequency components in signals with chaotic behavior

The analysis of chaotic signals with time-frequency methods is considered. For this purpose, two new transformations are presented which consist in the decomposition of a signal onto an orthogonal set of respectively linear and hyperbolic chirps. The linear chirp transformation is able to discriminate and extract particular chaotic components in non-stationary square integrable signals. This is demonstrated in an example studying the reflectometry measures of a turbulent plasma. The hyperbolic chirp transformation is designed for the detection and extraction of chaotic parts in self-similar processes such as stochastic motions. Mathematical connections are made between these two methods and other well-known transformations.

nlin.CD↗
arXiv2008

On the skeleton method and an application to a quantum scissor

In the spectral analysis of few one dimensional quantum particles interacting through delta potentials it is well known that one can recast the problem into the spectral analysis of an integral operator (the skeleton) living on the submanifold which supports the delta interactions. We shall present several tools which allow direct insight into the spectral structure of this skeleton. We shall illustrate the method on a model of a two dimensional quantum particle interacting with two infinitely long straight wires which cross one another at a certain angle : the quantum scissor.

math-ph↗
arXiv2006

Effective models for excitons in carbon nanotubes

We analyse the low lying spectrum of a model of excitons in carbon nanotubes. Consider two particles with an attractive Coulomb self-interaction, placed on an infinitely long cylinder. If the cylinder radius becomes small, the low lying spectrum is well described by a one-dimensional effective Hamiltonian which is exactly solvable.

math-ph↗