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Gabor frame bound optimizations

Markus Faulhuber, Irina Shafkulovska

TL;DR

This work analyzes extremal lattice choices for sharp Gabor frame bounds across several window families on rectangular lattices with integer density $n\ge2$. Using a combination of Zak-duality, dilation invariance, and meticulous interval estimates for hyperbolic-function expressions, it shows that optimal lattices can be highly window-dependent and, in some cases, do not exist as nondegenerate configurations. Notably, the hyperbolic secant window yields a unique global optimum at the square lattice, while cut-off, one-sided, and two-sided exponentials exhibit density- and-parameter–dependent optimals, with sometimes distinct lattices optimizing the lower bound, upper bound, and the condition number. Overall, the results highlight that Gaussian-like intuition does not generally carry over and that optimal lattice existence and location must be assessed separately for each window and density, revealing a rich landscape of lattice optimization in Gabor analysis.

Abstract

We study sharp frame bounds of Gabor systems over rectangular lattices for different windows and integer oversampling rate. In some cases we obtain optimality results for the square lattice, while in other cases the lattices optimizing the frame bounds and the condition number are rectangular lattices which are different for the respective quantities. Also, in some cases optimal lattices do not exist at all and a degenerated system is optimal.

Gabor frame bound optimizations

TL;DR

This work analyzes extremal lattice choices for sharp Gabor frame bounds across several window families on rectangular lattices with integer density . Using a combination of Zak-duality, dilation invariance, and meticulous interval estimates for hyperbolic-function expressions, it shows that optimal lattices can be highly window-dependent and, in some cases, do not exist as nondegenerate configurations. Notably, the hyperbolic secant window yields a unique global optimum at the square lattice, while cut-off, one-sided, and two-sided exponentials exhibit density- and-parameter–dependent optimals, with sometimes distinct lattices optimizing the lower bound, upper bound, and the condition number. Overall, the results highlight that Gaussian-like intuition does not generally carry over and that optimal lattice existence and location must be assessed separately for each window and density, revealing a rich landscape of lattice optimization in Gabor analysis.

Abstract

We study sharp frame bounds of Gabor systems over rectangular lattices for different windows and integer oversampling rate. In some cases we obtain optimality results for the square lattice, while in other cases the lattices optimizing the frame bounds and the condition number are rectangular lattices which are different for the respective quantities. Also, in some cases optimal lattices do not exist at all and a degenerated system is optimal.
Paper Structure (41 sections, 10 theorems, 156 equations, 10 figures)

This paper contains 41 sections, 10 theorems, 156 equations, 10 figures.

Table of Contents

  1. Introduction
  2. The results
  3. Hyperbolic secant
  4. Cut-off exponentials
  5. The case m=1
  6. The case m=2
  7. One-sided exponential
  8. Two-sided exponential
  9. Notation and preliminaries
  10. Preparations
  11. The hyperbolic functions
  12. Interval estimate
  13. The hyperbolic secant
  14. The lower bound
  15. The upper bound
  16. ...and 26 more sections

Key Result

Theorem 2.1

Let $g(t) = \left( \frac{\pi}{2} \right)^{1/2} \mathop{\mathrm{sech}}\nolimits(\pi t)$ be the normalized and Fourier-invariant hyperbolic secant and consider the rectangular lattice $\Lambda_{a, b} = a \mathbb{Z} \times b \mathbb{Z}$ of density $n$, i.e. $(a b)^{-1} = n$, with $2 \leq n \in \mathbb with equality if and only if $a = b = \tfrac{1}{\sqrt{n}}$, i.e., if the lattice is the square latt

Figures (10)

  • Figure 1: Estimate of the above type for $\tanh(t) (\mathop{\mathrm{csch}}\nolimits(t)-\mathop{\mathrm{csch}}\nolimits(2t))$ on the interval $(2,5)$. In order to get more appropriate estimates we may use the sub-intervals $(2,3)$ and $(3,5)$.
  • Figure 2: The re-normalized optimal bounds $n^{-1} A(\eta/n,1/n)$ and $n^{-1} B(\eta/n,1/n)$ for densities $n \in \{2,5,8\}$. As $n$ grows, the bounds become extremely flat around the optimum. For both cases the optimizing parameter $\eta$ (depending on $n$) yields a square lattice of density $n$.
  • Figure 3: Comparison of the behavior of $f_A$ and $f_B$ as well as $h_A$ and $h_B$.
  • Figure 4: Frame bounds of the Gabor systems $\mathcal{G}(g_\gamma, a \mathbb{Z} \times b \mathbb{Z})$ for $(ab)^{-1} = n \in \{1,3,6\}$ and $a=1, \, b = 1/n$ and $a = b = \sqrt{1/n}$. For $\gamma = 0$, we always have a tight frame with bounds $A=B=n$. The upper frame bounds (black) are increasing with $\gamma$ whereas the lower frame bounds (gray, dashed) are decreasing.
  • Figure 5: Lower frame bounds and upper frame bounds in dependence of the lattice parameter $a$ for $\gamma=1$ and $n \in \{1,3,6\}$. We see that $A$ has a unique maximum, whereas $B$ may not have a minimizing lattice (the lattice is degenerated for $a=0$).
  • ...and 5 more figures

Theorems & Definitions (12)

  • Theorem 2.1
  • Corollary 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Proposition 3.1
  • Lemma 5.1
  • proof
  • ...and 2 more