Coherence in randomness: On the zeros of some time-frequency transforms of white noise
vendredi 18 octobre 2019, 9h30 - 10h30
based on joint work with Julien Flamant, Pierre Chainais, and Adrien Hardy
Spectrograms are the musical scores of signal processing: A spectrogram is the modulus of a particular linear transform of the signal, and its values can be interpreted as measuring how much of the signal’s energy is spent on each time and frequency. Many signal processing algorithms thus rely on finding the maxima of a spectrogram in the time-frequency plane. In this talk, we take the dual approach of rather looking at the zeros of spectrograms, which seem to have more mathematical structure.
As first noted by Flandrin , when the linear transform used in the spectrogram is a short-time Fourier transform with a Gaussian window, the zeros of the spectrogram of white noise spread very uniformly over the time-frequency plane, as if these zeros « repelled » each other. This is akin to « antibunching » in optics. So much akin actually, that we initially wanted to prove that these zeros were a « determinantal point process », a probability distribution over sets of points that was first introduced in optics to model beams of fermions. We will see that for the short-time Fourier transform, the zeros of the spectrogram of white noise are not determinantal. However, we will give a general recipe to tweak the linear transform in the spectrogram for these zeros to span known point processes, including determinantal ones. In this process, we will recover some known time-frequency transforms, and find new strange ones.