Publication

Wavelet and footprint sampling of signals with a finite rate of innovation

Martin Vetterli, Pier Luigi Dragotti
2004
Article de conférence
Résumé

In this paper, we consider classes of not bandlimited signals, namely, streams of Diracs and piecewise polynomial signals, and show that these signals can be sampled and perfectly reconstructed using wavelets as sampling kernel. Due to the multiresolution structure of the wavelet transform, these new sampling theorems naturally lead to the development of a new resolution enhancement algo- rithm based on wavelet footprints [2]. Preliminary results show the potentiality of this algorithm.

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Concepts associés (22)
Wavelet transform
In mathematics, a wavelet series is a representation of a square-integrable (real- or complex-valued) function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal wavelet and of the integral wavelet transform. A function is called an orthonormal wavelet if it can be used to define a Hilbert basis, that is a complete orthonormal system, for the Hilbert space of square integrable functions.
Discrete wavelet transform
In numerical analysis and functional analysis, a discrete wavelet transform (DWT) is any wavelet transform for which the wavelets are discretely sampled. As with other wavelet transforms, a key advantage it has over Fourier transforms is temporal resolution: it captures both frequency and location information (location in time). Haar wavelet The first DWT was invented by Hungarian mathematician Alfréd Haar. For an input represented by a list of numbers, the Haar wavelet transform may be considered to pair up input values, storing the difference and passing the sum.
Ondelette
thumb|Ondelette de Daubechies d'ordre 2. Une ondelette est une fonction à la base de la décomposition en ondelettes, décomposition similaire à la transformée de Fourier à court terme, utilisée dans le traitement du signal. Elle correspond à l'idée intuitive d'une fonction correspondant à une petite oscillation, d'où son nom. Cependant, elle comporte deux différences majeures avec la transformée de Fourier à court terme : elle peut mettre en œuvre une base différente, non forcément sinusoïdale ; il existe une relation entre la largeur de l'enveloppe et la fréquence des oscillations : on effectue ainsi une homothétie de l'ondelette, et non seulement de l'oscillation.
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Publications associées (33)

Wavelet variance scale-dependence as a dynamics discriminating tool in high-frequency urban wind speed time series

Dasaraden Mauree, Fabian Guignard

High frequency wind time series measured at different heights from the ground (from 1.5 to 25.5 meters) in an urban area were investigated by using the variance of the coefficients of their wavelet transform. Two ranges of scales were identified, sensitive ...
2019

Wavelet variance scale-dependence as a dynamics discriminating tool in high-frequency urban wind speed time series

Dasaraden Mauree, Fabian Guignard

High frequency wind time series measured at different heights from the ground (from 1.5 to 25.5 meters) in an urban area were investigated by using the variance of the coefficients of their wavelet transform. Two ranges of scales were identified, sensitive ...
2018

Variational Justification of Cycle Spinning for Wavelet-Based Solutions of Inverse Problems

Michaël Unser, Emrah Bostan, Ulugbek Kamilov

Cycle spinning is a widely used approach for improving the performance of wavelet-based methods that solve linear inverse problems. Extensive numerical experiments have shown that it significantly improves the quality of the recovered signal without increa ...
Ieee-Inst Electrical Electronics Engineers Inc2014
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MOOCs associés (6)
Digital Signal Processing I
Basic signal processing concepts, Fourier analysis and filters. This module can be used as a starting point or a basic refresher in elementary DSP
Digital Signal Processing II
Adaptive signal processing, A/D and D/A. This module provides the basic tools for adaptive filtering and a solid mathematical framework for sampling and quantization
Digital Signal Processing III
Advanced topics: this module covers real-time audio processing (with examples on a hardware board), image processing and communication system design.
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