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The Analytic Stockwell Transform and its Zeros

Ali Moukadem, Barbara Pascal, Jean-Baptiste Courbot, Nicolas Juillet

TL;DR

This work introduces the Analytic Stockwell Transform (AST) by leveraging a modulated Cauchy wavelet, then proves the existence of an analytic representation on the unit disk via a Cayley transform and a nonvanishing factor. It characterizes the zeros of the AST of white noise by linking them, in law, to the zeros of the hyperbolic Gaussian analytic function, establishing invariance under the Poincaré disk isometries. Theoretical zero statistics are complemented by intensive Monte Carlo simulations comparing the empirical zero patterns with the hyperbolic GAF predictions, supported by a publicly available Python toolbox. The results underpin zero-based approaches for detection, denoising, and component separation in time–frequency analysis, with potential extensions to hyperbolic-adaptive localization and off-the-grid methods.

Abstract

A recent original line of research in time--frequency analysis has shifted the interest in energy maxima toward zeros. Initially motivated by the intriguing uniform spread of the zeros of the spectrogram of white noise, it has led to fruitful theoretical developments combining probability theory, complex analysis and signal processing. In this vein, the present work proposes a characterization of the zeros of the Stockwell Transform of white noise, which consists in an hybrid time--frequency multiresolution representation. First of all, an analytic version of the Stockwell Transform is designed. Then, analyticity is leveraged to establish a connection with the hyperbolic Gaussian Analytic Function, whose zero set is invariant under the isometries of the Poincaré disk. Finally, the theoretical spatial statistics of the zeros of the hyperbolic Gaussian Analytic Function and the empirical statistics of the zeros the Analytic Stockwell Transform of white noise are compared through intensive Monte Carlo simulations, supporting the established connection. A publicly available documented Python toolbox accompanies this work.

The Analytic Stockwell Transform and its Zeros

TL;DR

This work introduces the Analytic Stockwell Transform (AST) by leveraging a modulated Cauchy wavelet, then proves the existence of an analytic representation on the unit disk via a Cayley transform and a nonvanishing factor. It characterizes the zeros of the AST of white noise by linking them, in law, to the zeros of the hyperbolic Gaussian analytic function, establishing invariance under the Poincaré disk isometries. Theoretical zero statistics are complemented by intensive Monte Carlo simulations comparing the empirical zero patterns with the hyperbolic GAF predictions, supported by a publicly available Python toolbox. The results underpin zero-based approaches for detection, denoising, and component separation in time–frequency analysis, with potential extensions to hyperbolic-adaptive localization and off-the-grid methods.

Abstract

A recent original line of research in time--frequency analysis has shifted the interest in energy maxima toward zeros. Initially motivated by the intriguing uniform spread of the zeros of the spectrogram of white noise, it has led to fruitful theoretical developments combining probability theory, complex analysis and signal processing. In this vein, the present work proposes a characterization of the zeros of the Stockwell Transform of white noise, which consists in an hybrid time--frequency multiresolution representation. First of all, an analytic version of the Stockwell Transform is designed. Then, analyticity is leveraged to establish a connection with the hyperbolic Gaussian Analytic Function, whose zero set is invariant under the isometries of the Poincaré disk. Finally, the theoretical spatial statistics of the zeros of the hyperbolic Gaussian Analytic Function and the empirical statistics of the zeros the Analytic Stockwell Transform of white noise are compared through intensive Monte Carlo simulations, supporting the established connection. A publicly available documented Python toolbox accompanies this work.
Paper Structure (12 sections, 5 theorems, 50 equations, 3 figures)

This paper contains 12 sections, 5 theorems, 50 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Analytic Stockwell Transform
  3. Preliminary: definition of the Generalized Stockwell Transform
  4. Existence of an Analytic Stockwell Transform
  5. The Zeros of the Analytic Stockwell Transform of White Noise
  6. Gaussian analytic functions and point processes
  7. Link with the zeros of the hyperbolic Gaussian analytic function
  8. Numerical Experiments and Spatial Statistics of Zeros
  9. Discrete Time--Frequency Analysis
  10. Spatial Statistics
  11. Conclusion and Perspectives
  12. Acknowledgements

Key Result

Theorem 1

Let $\beta > 0$ and $\psi_{\beta} \in L^2(\mathbb{R})$ be the Cauchy wavelet of parameter $\beta$. Define the modulated Cauchy wavelet $\varphi_{\beta}(t) = \psi_{\beta}(t) \mathrm{e}^{-2\pi \mathrm{i} t}$. There exists a smooth nonvanishing function $\lambda : \mathbb{R} \times \mathbb{R}_+ \righta with $F : \mathbb{D} \rightarrow \mathbb{C}$ an analytic function on the unit disk.

Figures (3)

  • Figure 1: Log-modulus of the Analytic Stockwell Transform of white noise (background colormap) and its zero set (red dots) $\mathfrak{Z}^{(\alpha)}$. The discrete white noise is considered to span a time period of $x_{\max} - x_{\min} = 10$ s with a sampling frequency of $400$ Hz. The parameter of the Cauchy wavelet $\beta$ is chosen such that $\alpha=2\beta+1=300$. (Left) Representation in the $(x, \xi^{-1})$-plane where the inverse frequency $\xi^{-1}$ is called the scale in reference to the related Cauchy Wavelet Transform. (Right) Representation in the Poincaré disk $\mathbb{D}$ parameterized by $\vartheta(x + \mathrm{i} \xi^{-1})$ for $(x,\xi^{-1})$ running over the same grid, where $\vartheta$ is the Cayley transform of Equation \ref{['eq:Cayley']}. The dotted circle represents the border of $\mathbb{D}$.
  • Figure 2: Key features of the implementation of the pair correlation function estimator $\widehat{g}^{(\alpha)}$: counting zeros in hyperbolic disks (left) and border effect correction (right).
  • Figure 3: Comparison of theoretical and empirical spatial statistics of the zeroes of the Analytic Stockwell Transform of parameter $\beta = (\alpha-1)/2$ of white noise for $\alpha \in \lbrace 50, 100, 300, 500\rbrace$. (Left) First intensity $\rho_1^{(\alpha)}$. (Right) pair correlation function $g^{(\alpha)}$. Theoretical spatial statistics, \ref{['eq:rho1']} and \ref{['eq:g_th']}, are represented as dashed black lines and averaged empirical statistics, \ref{['eq:hatrho1']} and \ref{['eq:hatgalpha']}, as solid blue lines. Ensemble averaging and quantiles estimation are performed over $R = 100$ realizations of the discrete white noise of $N= 4000$ points, corresponding to a time period of $x_{\max} - x_{\min} = 1$ s long and a sampling frequency $\nu_s = 4000$ Hz. The $x$-axis is expressed in pseudo-hyperbolic distances \ref{['eq:dph_disk']}.

Theorems & Definitions (24)

  • Definition 1
  • Remark 1
  • Definition 2
  • Remark 2
  • Theorem 1
  • proof
  • Definition 3
  • Definition 4
  • Theorem 2
  • Lemma 1
  • ...and 14 more