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Operator-isomorphism pairs and Zak transform methods for the study of Gabor systems

Markus Faulhuber

TL;DR

The paper investigates when Gabor systems $\mathcal{G}(g,\Lambda)$ are unitarily equivalent via operator-isomorphism pairs, with a focus on integer over-sampling and Hermite windows. It develops a framework combining intertwinings of time-frequency shifts, Zak transform techniques, and the fractional Fourier transform to transfer frame questions across equivalent systems. New proofs and non-frame results for Hermite functions are presented (e.g., for $h_{4\ell+2}$ and $h_2$ on specific lattices), together with an outlook on double over-sampling as a path to frames. The work provides a symmetry-based toolkit to analyze and potentially construct Gabor frames by translating problems into Zak-domain zeros and lattice transformations, guiding over-sampling strategies in signal processing.

Abstract

We collect and summarize results on the unitary equivalence of Gabor systems by pairs of unitary operators and global isometries. The methods are then used to study Gabor systems with Hermite functions. We provide new proofs of some known results and an outlook on double over-sampling.

Operator-isomorphism pairs and Zak transform methods for the study of Gabor systems

TL;DR

The paper investigates when Gabor systems are unitarily equivalent via operator-isomorphism pairs, with a focus on integer over-sampling and Hermite windows. It develops a framework combining intertwinings of time-frequency shifts, Zak transform techniques, and the fractional Fourier transform to transfer frame questions across equivalent systems. New proofs and non-frame results for Hermite functions are presented (e.g., for and on specific lattices), together with an outlook on double over-sampling as a path to frames. The work provides a symmetry-based toolkit to analyze and potentially construct Gabor frames by translating problems into Zak-domain zeros and lattice transformations, guiding over-sampling strategies in signal processing.

Abstract

We collect and summarize results on the unitary equivalence of Gabor systems by pairs of unitary operators and global isometries. The methods are then used to study Gabor systems with Hermite functions. We provide new proofs of some known results and an outlook on double over-sampling.

Paper Structure

This paper contains 9 sections, 3 theorems, 56 equations, 3 figures.

Table of Contents

  1. Introduction and Notation
  2. Intertwining of time-frequency shifts
  3. Valid Operator-isomorphism pairs
  4. The Zak transform
  5. Properties of the Zak transform
  6. The Zak transform and Hermite functions
  7. An alternative proof of Lemvig's result
  8. Another proof that G (h2, sqrt(1/2) Z 2) is not a frame
  9. Gabor frames: an outlook

Key Result

Lemma 6.1

The Hermite function $h_n$ is an eigenfunction of the fractional Fourier transform $\mathcal{F}_r$ with eigenvalue $e^{- i n r}$:

Figures (3)

  • Figure 1: Surface plot of $|\mathcal{Z} \mathcal{D}_{\sqrt{2}}^{-1} h_2(x,\omega)|$ with contour lines on the bottom. $\mathcal{Z} \mathcal{D}_{\sqrt{2}}^{-1} h_2(x,\omega)$ has zeros located at $(x,\omega) \in \{(\frac{1}{4}, \frac{1}{2}), (\frac{1}{2}, \frac{1}{2}), (\frac{3}{4}, \frac{1}{2})\} + \mathbb{Z}^2$. Note that numerically $\mathcal{Z} \mathcal{D}_{\sqrt{2}}^{-1} h_2 (0,\omega) \neq 0$ and $\mathcal{Z} \mathcal{D}_{\sqrt{2}}^{-1} h_2 (x,0) \neq 0$.
  • Figure 2: Decomposition of the square lattice $\frac{1}{\sqrt{2}} \mathbb{Z}^2$ into the rectangular lattice $D_{\sqrt{2}} \mathbb{Z}^2$, which has a side ratio of 2, and the shifted copy $D_{\sqrt{2}}(\mathbb{Z}+\frac{1}{2})^2$.
  • Figure 3: Surface plot of $|\mathcal{Z} h_2(x,\omega)|$ with contour lines on the bottom. Zeros of $\mathcal{Z} h_2(x,\omega)$ are located at $(x,\omega) \in \mathbb{Z}^2 \cup \left(\mathbb{Z} + \frac{1}{2}\right)^2$.

Theorems & Definitions (3)

  • Lemma 6.1
  • Corollary 6.2
  • Proposition 6.3