On the zeros of the spectrogram of white noise

Rémi Bardenet (1, 2) , Julien Flamant (2) , Pierre Chainais (2, 3)

1 CNRS - Centre National de la Recherche Scientifique
2 CRIStAL - Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189
3 Centrale Lille

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 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 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].

Domaines: Statistiques [math.ST]; Traitement du signal et de l'image [eess.SP]

Rémi Bardenet, Julien Flamant, Pierre Chainais. On the zeros of the spectrogram of white noise. Applied and Computational Harmonic Analysis, 2020, 48 (2), pp.682-705. ⟨10.1016/j.acha.2018.09.002⟩. ⟨hal-01572207⟩