Titre :

Time-frequency transforms of white noises and Gaussian analytic functions

Auteur(s) :

Bardenet, Remi [Auteur]
Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 [CRIStAL]
Centre National de la Recherche Scientifique [CNRS]
Hardy, Adrien [Auteur]
Méthodes quantitatives pour les modèles aléatoires de la physique [MEPHYSTO-POST]
Laboratoire Paul Painlevé - UMR 8524 [LPP]

Titre de la revue :

Applied and Computational Harmonic Analysis

Éditeur :

Elsevier

Date de publication :

2019

ISSN :

1063-5203

Mot(s)-clé(s) en anglais :

Determinantal point process
Signal detection and reconstruction
Short-time Fourier transform
Gaussian analytic functions

Discipline(s) HAL :

Mathématiques [math]/Statistiques [math.ST]
Mathématiques [math]/Probabilités [math.PR]
Sciences de l'ingénieur [physics]/Traitement du signal et de l'image [eess.SP]

Résumé en anglais : [en]

A family of Gaussian analytic functions (GAFs) has recently been linked to the Gabor transform of white Gaussian noise [Bardenet et al., 2017]. This answered pioneering work by Flandrin [2015], who observed that the zeros ...
Lire la suite >A family of Gaussian analytic functions (GAFs) has recently been linked to the Gabor transform of white Gaussian noise [Bardenet et al., 2017]. This answered pioneering work by Flandrin [2015], who observed that the zeros of the Gabor transform of white noise had a very regular distribution and proposed filtering algorithms based on the zeros of a spectrogram. The mathematical link with GAFs provides a wealth of probabilistic results to inform the design of such signal processing procedures. In this paper, we study in a systematic way the link between GAFs and a class of time-frequency transforms of Gaussian white noises on Hilbert spaces of signals. Our main observation is a conceptual correspondence between pairs (transform, GAF) and generating functions for classical orthogonal polynomials. This correspondence covers some classical time-frequency transforms, such as the Gabor transform and the Daubechies-Paul analytic wavelet transform. It also unveils new windowed discrete Fourier transforms, which map white noises to fundamental GAFs. All these transforms may thus be of interest to the research program " filtering with zeros ". We also identify the GAF whose zeros are the extrema of the Gabor transform of the white noise and derive their first intensity. Moreover, we discuss important subtleties in defining a white noise and its transform on infinite dimensional Hilbert spaces. Finally, we provide quantitative estimates concerning the finite-dimensional approximations of these white noises, which is of practical interest when it comes to implementing signal processing algorithms based on GAFs.Lire moins >

Langue :

Anglais

Comité de lecture :

Oui

Audience :

Internationale

Vulgarisation :

Non

Projet ANR :

Inférence bayésienne à ressources limitées - données massives et modèles coûteux

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL