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Empirical plunge profiles of time-frequency localization operators

Simon Halvdansson

TL;DR

The paper analyzes the eigenvalue decay (plunge region) of time-frequency localization operators as the dilation parameter R grows. For rotationally invariant symbols (disks, annuli, unions), it derives explicit large-R asymptotics with an erfc plunge profile and shows that the same form extends universally to general Ω under a Gaussian window, up to O(1/R) errors. Numerical validation using discrete frame multipliers in LTFAT across various shapes supports the erfc predictions and highlights the influence of symbol geometry and window choice on the plunge behavior. The work connects to universality phenomena in probability and random matrix theory, suggesting that boundary-localized eigenfunctions govern the plunge region and that the erfc form captures the essential asymptotics of time-frequency localization gaps in the large-R limit.

Abstract

For time-frequency localization operators, related to the short-time Fourier transform, with symbol $RΩ$, we work out the exact large $R$ eigenvalue behavior for rotationally invariant $Ω$ and conjecture that the same relation holds for all scaled symbols $R Ω$ as long as the window is the standard Gaussian. Specifically, we conjecture that the $k$-th eigenvalue of the localization operator with symbol $RΩ$ converges to $\frac{1}{2}\operatorname{erfc}\big( \sqrt{2π}\frac{k-R^2|Ω|}{R|\partial Ω|} \big)$ 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.

Empirical plunge profiles of time-frequency localization operators

TL;DR

The paper analyzes the eigenvalue decay (plunge region) of time-frequency localization operators as the dilation parameter R grows. For rotationally invariant symbols (disks, annuli, unions), it derives explicit large-R asymptotics with an erfc plunge profile and shows that the same form extends universally to general Ω under a Gaussian window, up to O(1/R) errors. Numerical validation using discrete frame multipliers in LTFAT across various shapes supports the erfc predictions and highlights the influence of symbol geometry and window choice on the plunge behavior. The work connects to universality phenomena in probability and random matrix theory, suggesting that boundary-localized eigenfunctions govern the plunge region and that the erfc form captures the essential asymptotics of time-frequency localization gaps in the large-R limit.

Abstract

For time-frequency localization operators, related to the short-time Fourier transform, with symbol , we work out the exact large eigenvalue behavior for rotationally invariant and conjecture that the same relation holds for all scaled symbols as long as the window is the standard Gaussian. Specifically, we conjecture that the -th eigenvalue of the localization operator with symbol converges to as . 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.

Paper Structure

This paper contains 10 sections, 5 theorems, 69 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction and background
  2. Earlier results
  3. Our contribution
  4. Eigenvalue behavior
  5. Eigenvalues on disks, annuli, and rotationally invariant sets
  6. Universality
  7. Numerical verification
  8. Setup
  9. Symbol and frame dependence
  10. Window dependence

Key Result

Theorem 2.1

Let $\lambda_k^R$ be the $k$-th eigenvalue of the localization operator $A_{B(0,R)}^{g_0}$. It then holds that where $\operatorname{erfc}$ is the complementary error function.

Figures (12)

  • Figure 1: Numerical verification of Proposition \ref{['prop:loc_op_annulus_eig']} with $R = 15$, $r = 0.6$ comparing a manual sorting of samples of $f$ with the proposed $\Tilde{f}^*$.
  • Figure 2: Experiment with $\textsf{a} = 10, \textsf{M} = 100$, Gaussian window and a disk as the symbol. Maximum error is close to $1.0$%.
  • Figure 3: Experiment with $\textsf{a} = 10, \textsf{M} = 200$, Gaussian window and a disk as the symbol. Maximum error is close to $0.8$%.
  • Figure 4: Experiment with $\textsf{a} = 10, \textsf{M} = 100$, Gaussian window and a star shape as the symbol. Maximum error is close to $0.7$%.
  • Figure 5: Experiment with $\textsf{a} = 5, \textsf{M} = 40$, Gaussian window and a star shape as the symbol. Maximum error is close to $1.6$%.
  • ...and 7 more figures

Theorems & Definitions (14)

  • Theorem 2.1
  • proof
  • Remark
  • Remark
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 4 more