On the zeros of the spectrogram of white noise

Bardenet, Rémi; Flamant, Julien; Chainais, Pierre

doi:10.1016/j.acha.2018.09.002

Cited by 29 publications

(60 citation statements)

References 28 publications

(33 reference statements)

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“…Theorem 2.1 has already been obtained in [Bardenet et al, 2017] with a slightly different definition of the white noise, taking its values in the space of tempered distributions. We provide in Section 4.1 an alternative proof which fits in our more general setting.…”

Section: The Planar Gafmentioning

confidence: 99%

“…[Holden et al, 2010, Section 2.1]. This is the approach followed by Bardenet et al [2017] to identify the Gabor transform of white noise. However, the search for "smaller" abstract Wiener spaces like Θ is philosophically interesting: since elements outside of H are usually hard to interpret physically, it is desirable to add as few elements as possible to the space H. Let us now describe the support Θ of the white noise used in this work in several examples and observe that it is smaller than S (T, C).…”

Section: Support Of the White Noisementioning

confidence: 99%

“…This provided a first link between zeros of random spectrograms and GAFs, which explained the observations of Flandrin [2015]. Bardenet et al [2017] then used probabilistic results on this planar GAF to inform the design of signal reconstruction procedures in the spirit of [Flandrin, 2015].…”

mentioning

confidence: 99%

“…We answer here this question affirmatively. Conversely, since the GAF identified by Bardenet et al [2017] is one of the three with special invariance properties, a probabilist may ask whether the other two also naturally appear. We also give here an affirmative answer.…”

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confidence: 99%

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Time-frequency transforms of white noises and Gaussian analytic functions

Bardenet

Hardy

2021

Applied and Computational Harmonic Analysis

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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.

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Section: The Planar Gafmentioning

confidence: 99%

Section: Support Of the White Noisementioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Time-frequency transforms of white noises and Gaussian analytic functions

Bardenet

Hardy

2021

Applied and Computational Harmonic Analysis

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“…We were slightly frustrated of not finding a well-known DPP behind the STFT of white noise in [1]. In particular, for α > −1, consider the random analytic function…”

Section: From a Dpp To Analytic Waveletsmentioning

confidence: 99%

A correspondence between zeros of time-frequency transforms and Gaussian analytic functions

Bardenet

Chainais

Flamant

et al. 2019

2019 13th International Conference on Sampling Theory and Applications (SampTA)

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In this paper, we survey our joint work on the point processes formed by the zeros of time-frequency transforms of Gaussian white noises [1], [2]. Unlike both references, we present the work from the bottom up, stating results in the order they came to us and commenting what we were trying to achieve. The route to our more general results in [2] was a sort of ping pong game between signal processing, harmonic analysis, and probability. We hope that narrating this game gives additional insight into the more technical aspects of the two references. We conclude with a number of open problems that we believe are relevant to the SampTA community.

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Local Maxima of White Noise Spectrograms and Gaussian Entire Functions

Abreu

2022

J Fourier Anal Appl

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We confirm Flandrin’s prediction for the expected average of local maxima of spectrograms of complex white noise with Gaussian windows (Gaussian spectrograms or, equivalently, modulus of weighted Gaussian Entire Functions), a consequence of the conjectured double honeycomb mean model for the network of zeros and local maxima, where the area of local maxima centered hexagons is three times larger than the area of zero centered hexagons. More precisely, we show that Gaussian spectrograms, normalized such that their expected density of zeros is 1, have an expected density of 5/3 critical points, among those 1/3 are local maxima, and 4/3 saddle points, and compute the distributions of ordinate values (heights) for spectrogram local extrema. This is done by first writing the spectrograms in terms of Gaussian Entire Functions (GEFs). The extrema are considered under the translation invariant derivative of the Fock space (which in this case coincides with the Chern connection from complex differential geometry). We also observe that the critical points of a GEF are precisely the zeros of a Gaussian random function in the first higher Landau level. We discuss natural extensions of these Gaussian random functions: Gaussian Weyl–Heisenberg functions and Gaussian bi-entire functions. The paper also reviews recent results on the applications of white noise spectrograms, connections between several developments, and is partially intended as a pedestrian introduction to the topic.

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On the zeros of the spectrogram of white noise

Cited by 29 publications

References 28 publications

Time-frequency transforms of white noises and Gaussian analytic functions

Time-frequency transforms of white noises and Gaussian analytic functions

A correspondence between zeros of time-frequency transforms and Gaussian analytic functions

Local Maxima of White Noise Spectrograms and Gaussian Entire Functions

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