On accumulated spectrograms

TL;DR

This paper analyzes the eigenvalues and eigenfunctions of time-frequency localization operators, showing that accumulated spectrograms of large eigenvalue eigenfunctions approximate the domain's measure and can be used for domain reconstruction.

Contribution

It introduces the concept of accumulated spectrograms for arbitrary domains and provides error estimates, enabling domain approximation solely from spectrograms without phase information.

Findings

Accumulated spectrograms form an approximate partition of unity over the domain.

Eigenvalues near 1 correspond to the measure of the domain with controlled error.

Spectrograms can be used to reconstruct the domain without phase data.

Abstract

We study the eigenvalues and eigenfunctions of the time-frequency localization operator $H_\Omega$ on a domain $\Omega$ of the time-frequency plane. The eigenfunctions are the appropriate prolate spheroidal functions for an arbitrary domain $\Omega$. Indeed, in analogy to the classical theory of Landau-Slepian-Pollak, the number of eigenvalues of $H_\Omega$ in $[1-\delta , 1]$ is equal to the measure of $\Omega$ up to an error term depending on the perimeter of the boundary of $\Omega$. Our main results show that the spectrograms of the eigenfunctions corresponding to the large eigenvalues (which we call the accumulated spectrogram) form an approximate partition ofunity of the given domain $\Omega $. We derive both asymptotic, non-asymptotic, and weak $L^2$ error estimates for the accumulated spectrogram. As a consequence the domain $\Omega$ can be approximated solely from the spectrograms of eigenfunctions without information about their phase.