The Analytic Stockwell Transform and its Zeros

TL;DR

This paper characterizes the zeros of the Stockwell Transform of white noise, linking it to hyperbolic Gaussian Analytic Functions, and validates findings through simulations and a Python toolbox.

Contribution

It introduces an analytic version of the Stockwell Transform and establishes a theoretical connection with hyperbolic Gaussian Analytic Functions for the first time.

Findings

Zeros of the Stockwell Transform of white noise follow a hyperbolic Gaussian distribution.

The empirical zero distribution matches the theoretical predictions from hyperbolic Gaussian Analytic Functions.

A Python toolbox is provided for practical analysis and replication.

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 Poincar\'e 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.