On the zeros of the spectrogram of white noise

TL;DR

This paper rigorously characterizes the distribution of zeros in the spectrogram of white Gaussian noise, providing a mathematical foundation for zero-based signal detection methods.

Contribution

It formally defines and analyzes the zeros of the spectrogram of white Gaussian noise, linking them to Gaussian analytic functions and exploring their use in signal detection.

Findings

Zeros are uniformly spread over the time-frequency plane.

Zeros of white Gaussian noise spectrograms correspond to Gaussian analytic functions.

Provides statistical and computational insights for zero-based signal detection.

Abstract

In a recent paper, Flandrin [2015] has proposed filtering based on the zeros of a spectrogram, using the short-time Fourier transform and a Gaussian window. His results are based on empirical observations on the distribution of the zeros of the spectrogram of white Gaussian noise. These zeros tend to be uniformly spread over the time-frequency plane, and not to clutter. Our contributions are threefold: we rigorously define the zeros of the spectrogram of continuous white Gaussian noise, we explicitly characterize their statistical distribution, and we investigate the computational and statistical underpinnings of the practical implementation of signal detection based on the statistics of spectrogram zeros. In particular, we stress that the zeros of spectrograms of white Gaussian noise correspond to zeros of Gaussian analytic functions, a topic of recent independent mathematical interest [Hough et al., 2009].