Fractional Fourier transform

In mathematics, in the area of harmonic analysis, the fractional Fourier transform (FRFT) is a family of linear transformations generalizing the Fourier transform. It can be thought of as the Fourier transform to the n-th power, where n need not be an integer — thus, it can transform a function to any intermediate domain between time and frequency. Its applications range from filter design and signal analysis to phase retrieval and pattern recognition. The FRFT can be used to define fractional convolution, correlation, and other operations, and can also be further generalized into the linear canonical transformation (LCT). An early definition of the FRFT was introduced by Condon, by solving for the Green's function for phase-space rotations, and also by Namias, generalizing work of Wiener on Hermite polynomials. However, it was not widely recognized in signal processing until it was independently reintroduced around 1993 by several groups. Since then, there has been a surge

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Shift-Invariant and Sampling Spaces Associated With the Fractional Fourier Transform Domain

Ayush Bhandari

Shift-invariant spaces play an important role in sampling theory, multiresolution analysis, and many other areas of signal and image processing. A special class of the shift-invariant spaces is the class of sampling spaces in which functions are determined by their values on a discrete set of points. One of the vital tools used in the study of sampling spaces is the Zak transform. The Zak transform is also related to the Poisson summation formula and a common thread between all these notions is the Fourier transform.

2012

High-Order Accurate Local Schemes for Fractional Differential Equations

Daniel Henri Baffet, Jan Sickmann Hesthaven

High-order methods inspired by the multi-step Adams methods are proposed for systems of fractional differential equations. The schemes are based on an expansion in a weighted space. To obtain the schemes this expansion is terminated after terms. We study the local truncation error and its behavior with respect to the step-size h and P. Building on this analysis, we develop an error indicator based on the Milne device. Methods with fixed and variable step-size are tested numerically on a number of problems, including problems with known solutions, and a fractional version on the Van der Pol equation.

Springer Verlag2017

Efficiently Learning Fourier Sparse Set Functions

Mikhail Kapralov, Andreas Krause, Amir Zandieh

Learning set functions is a key challenge arising in many domains, ranging from sketching graphs to black-box optimization with discrete parameters. In this paper we consider the problem of efficiently learning set functions that are defined over a ground set of size n and that are sparse (say k-sparse) in the Fourier domain. This is a wide class, that includes graph and hypergraph cut functions, decision trees and more. Our central contribution is the first algorithm that allows learning functions whose Fourier support only contains low degree (say degree d = o(n)) polynomials using O(kd log n) sample complexity and runtime O(kn log(2) k log n log d). This implies that sparse graphs with k edges can, for the first time, be learned from O(k log n) observations of cut values and in linear time in the number of vertices. Our algorithm can also efficiently learn (sums of) decision trees of small depth. The algorithm exploits techniques from the sparse Fourier transform literature and is easily implementable. Lastly, we also develop an efficient robust version of our algorithm and prove l(2)/l(2) approximation guarantees without any statistical assumptions on the noise.

NEURAL INFORMATION PROCESSING SYSTEMS (NIPS)2019

MICRO-310(b): Signals and systems I (for SV)

Présentation des concepts et des outils de base pour l'analyse et la caractérisation des signaux, la conception de systèmes de traitement et la modélisation linéaire de systèmes pour les étudiants en sciences de la vie. Application de ces techniques au traitement et à la transmission de signaux.

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Présentation des concepts et des outils de base pour la caractérisation des signaux ainsi que pour l'analyse et la synthèse des systèmes linéaires (filtres ou canaux de transmission). Application de ces techniques au traitement du signal et aux télécommunications.

EE-205: Signals and systems (for EL&IC)

This class teaches the theory of linear time-invariant (LTI) systems. These systems serve both as models of physical reality (such as the wireless channel) and as engineered systems (such as electrical circuits, filters and control strategies).

Fourier transform

In physics and mathematics, the Fourier transform (FT) is a transform that converts a function into a form that describes the frequencies present in the original function. The output of the transfo

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,

Discrete Fourier transform

In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time F