The Analytic Stockwell Transform and its zeros

Abstract

The Stockwell Transform is a time–frequency representation resulting from an hybridization between the Short-Time Fourier Transform and the Continuous Wavelet Transform. Instead of focusing on energy maxima, an unorthodox line of research has recently shed the light on the zeros of time–frequency transforms, leading to fruitful theoretical developments combining probability theory, complex analysis and signal processing. While the distributions of zeros of the Short-Time Fourier Transform and of the Continuous Wavelet Transform of white noise have been precisely characterized, that of the Stockwell Transform of white noise zeros remains unexplored. To fill this gap, the present work proposes a characterization of the distribution of zeros of the Stockwell Transform of white noise taking advantage of a novel generalized Analytic Stockwell Transform. 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, showing the practical relevance of the established connection. A documented Python toolbox has been made publicly available by the authors.