A Note on Reassigned Gabor Spectrograms of Hermite Functions
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A Note on Reassigned Gabor Spectrograms of Hermite Functions Patrick Flandrin
Received: 25 May 2012 / Revised: 12 June 2012 / Published online: 4 December 2012 © Springer Science+Business Media New York 2012
Abstract An explicit form is given for the reassigned Gabor spectrogram of an Hermite function of arbitrary order. It is shown that the energy concentration sharply localizes outside the border of a clearance area limited by the “classical” circle where the Gabor spectrogram attains its maximum value, with a perfect localization that can only be achieved in the limit of infinite order. Keywords Gabor transform · Hermite functions · Time-frequency analysis · Reassignment Mathematics Subject Classification 42C40 · 42A38 · 94A12 1 Motivation 1.1 Why Hermite functions? Hermite functions form a family of orthonormal functions that are important and useful in many respects. One can first mention that they are eigenfunctions of the Fourier transform [7], thus generalizing the well-known relationship which holds for the Gaussian function, the latter happening to be precisely the first Hermite function. As a consequence, Hermite functions are also naturally encountered in a variety of “uncertainty” questions constraining a function and its Fourier transform. They are in particular eigenfunctions of localization operators in the time-frequency plane, with respect to either elliptic indicator functions in the Wigner representation Communicated by Hans G. Feichtinger. P. Flandrin () École Normale Supérieure de Lyon, Laboratoire de Physique (UMR 5672 CNRS), 46 allée d’Italie, 69364 Lyon Cedex 07, France e-mail: [email protected]
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J Fourier Anal Appl (2013) 19:285–295
[8, 9] or Gaussian-shaped weightings applied to Gabor transforms [6]. This makes them particularly useful in the context of multitaper time-frequency analysis [3, 16]. From a more physical perspective, Hermite functions are stationary eigenstates of the quantum harmonic oscillator [5], with phase-space diagrams that restrict to circles p 2 + q 2 = C in the classical limit. 1.2 Why Reassigned Gabor Spectrograms? This note is concerned with Hermite functions considered as functions of the time variable, and its purpose is to investigate their localization properties in the timefrequency plane. This will be done within the specific framework of Gabor spectrograms, because they are positive energy distributions with minimum spread [14]. This will involve furthermore a reassignment step [10, 15] so as to increase localization. It is known in fact [10] that reassignment allows for a perfect localization in the case of unimodular linear chirps defined as c(t) = ei(αt
2 +βt+γ )
;
(α, β, γ ∈ R),
(1)
and one of the questions to be answered in this note—suggested by observations reported, e.g., in [4], Chap. 1, Fig. 1.6—is to establish whether such a perfect localization is attained for Hermite functions too (it will be shown in Sect. 3.3 that this is true only in some asymptotical sense).
2 Hermite, Wigner and Gabor We consider orthonormal Hermite functi
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