Empirical plunge profiles of time-frequency localization operators

Abstract

For time-frequency localization operators, related to the short-time Fourier transform, with symbol $R\Omega$, we work out the exact large $R$ eigenvalue behavior for rotationally invariant $\Omega$ and conjecture that the same relation holds for all scaled symbols $R\Omega$ as long as the window is the standard Gaussian. Specifically, we conjecture that the $k$-th eigenvalue of the localization operator with symbol $R\Omega$ converges to $\frac{1}{2}\mathrm{erfc}\left(\sqrt{2\pi }\frac{k-R^2|\Omega |}{R|\partial \Omega |}\right)$ as $R\to \infty$. To support the conjecture, we compute the eigenvalues of discrete frame multipliers with various symbols using LTFAT and find that they agree with the behavior of the conjecture to a large degree.

20 pages, 12 figures

References in corpus (6)

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• An Inverse Problem for Localization Operators
• The Weyl-Heisenberg ensemble: hyperuniformity and higher Landau levels
• Spectral deviation of concentration operators for the short-time Fourier transform
• Measure-operator convolutions and applications to mixed-state Gabor multipliers
• On accumulated spectrograms for Gabor frames