Time-frequency transforms of white noises and Gaussian analytic functions

Abstract

A family of Gaussian analytic functions (GAFs) has recently been linked to the Gabor transform of Gaussian white noises [4] . This answered pioneering work by Flandrin [10] , who observed that the zeros of the Gabor transform of white noise had a regular distribution and proposed filtering algorithms based on the zeros of a spectrogram. In this paper, we study in a systematic way the link between GAFs and a class of time-frequency transforms of Gaussian white noises. Our main observation is a correspondence between pairs (transform, GAF) and generating functions for classical orthogonal polynomials. This covers some classical time-frequency transforms, such as the Gabor transform and the Daubechies-Paul wavelet transform. It also unveils new windowed discrete Fourier transforms, which map white noises to fundamental GAFs. Moreover, we discuss subtleties in defining a white noise and its transform on infinite dimensional Hilbert spaces and its finite dimensional approximations.