Companion repository to "Empirical plunge profiles of time-frequency localization operators"

This repository contains code to repeat the main experiments of the paper Empirical plunge profiles of time-frequency localization operators. Inside the LTFAT folder, plunge_profile.m contains the code for the main figures in the paper. Change the load_symbol argument to change the symbol. The code for the RGB figure can be found in acc_spec_color.m file. For the figure comparing different windows, use the acc_spec_color.m routine.

The python file f(k)_plot.py contains a numerical verification of Proposition 2.2 which was used to produce Figure 1 in the paper.

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 $\frac{1}{2}erfc\big( \sqrt{2\pi}\frac{k-R^2|\Omega|}{R|\partial \Omega|} \big)$ 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.

Sorted eigenvalues validation

Window dependence on eigenvalue spectrogram separation