On the Zeros of the Stockwell and Morlet Wavelet Transforms

Sur les zéros de la transformée de Stockwell et la transformée en ondelettes de Morlet

Ali Moukadem (1, 2) , Jean-Baptiste Courbot (1, 2) , Bruno Colicchio (1, 2) , Alain Dieterlen (1, 2)

1 IRIMAS - Institut de Recherche en Informatique Mathématiques Automatique Signal - IRIMAS - UR 7499
2 Université de Haute-Alsace (UHA) - Université de Haute-Alsace (UHA) Mulhouse - Colmar

Recently a link between Gaussian analytic functions (GAFs) and time-frequency transforms has been established by Bardenet et al. This work was motivated by the earlier work performed by Flandrin on the zeros of the Spectrogram (squared modulus of the short-time Fourier transform) and their regular distribution which form a point process in the time-frequency plane. The aim of this paper is to extend these earlier studies to the Stockwell transform (ST) which is an hybrid transform between the Short Time Fourier Transform (STFT) and the Continuous Wavelet Transform (CWT). First, the factorization of a generalized form of the ST in term of Bargmann transform is given. Then the ST of white Gaussian noise is developed formally in order to give the link between ST and planar GAFs. Because of the close relation between the ST and the Morlet Wavelet transform the zeros of this specific CWT are also discussed in this study. Finally, examples of zeros detected from the ST domain is illustrated on synthetic and real signals.

Domaines: Traitement du signal et de l'image [eess.SP]; Physique mathématique [math-ph]

Ali Moukadem, Jean-Baptiste Courbot, Bruno Colicchio, Alain Dieterlen. On the Zeros of the Stockwell and Morlet Wavelet Transforms. 2021. ⟨hal-03256836⟩