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Characterization and computation of canonical tight windows for Gabor frames

A. J. E. M Janssen, Thomas Strohmer

TL;DR

The paper addresses finding a canonical tight Gabor frame window that remains as close as possible to a given window. It proves, using time, frequency, time-frequency, and Zak-transform representations plus functional calculus, that the canonical tight window $h^{0}=S^{-1/2} g$ minimizes the distance to $g$ within the class of normalized tight frames; it also extends these ideas to a Wiener-Levy framework for rational oversampling. A practical Newton-type method is developed to compute $h^{0}$ efficiently, with quadratic convergence and guidance on optimal scaling, supported by convergence analysis and numerical experiments. The work provides a unified, domain-spanning treatment of tight Gabor frames and delivers a robust algorithm for tight-frame design that preserves the original window's characteristics, with implications for signal processing and communications under oversampling.

Abstract

Let $(g_{nm})_{n,m\in Z}$ be a Gabor frame for $L_2(R)$ for given window $g$. We show that the window $h^0=S^{-1/2} g$ that generates the canonically associated tight Gabor frame minimizes $\|g-h\|$ among all windows $h$ generating a normalized tight Gabor frame. We present and prove versions of this result in the time domain, the frequency domain, the time-frequency domain, and the Zak transform domain, where in each domain the canonical $h^0$ is expressed using functional calculus for Gabor frame operators. Furthermore, we derive a Wiener-Levy type theorem for rationally oversampled Gabor frames. Finally, a Newton-type method for a fast numerical calculation of $\ho$ is presented. We analyze the convergence behavior of this method and demonstrate the efficiency of the proposed algorithm by some numerical examples.

Characterization and computation of canonical tight windows for Gabor frames

TL;DR

The paper addresses finding a canonical tight Gabor frame window that remains as close as possible to a given window. It proves, using time, frequency, time-frequency, and Zak-transform representations plus functional calculus, that the canonical tight window minimizes the distance to within the class of normalized tight frames; it also extends these ideas to a Wiener-Levy framework for rational oversampling. A practical Newton-type method is developed to compute efficiently, with quadratic convergence and guidance on optimal scaling, supported by convergence analysis and numerical experiments. The work provides a unified, domain-spanning treatment of tight Gabor frames and delivers a robust algorithm for tight-frame design that preserves the original window's characteristics, with implications for signal processing and communications under oversampling.

Abstract

Let be a Gabor frame for for given window . We show that the window that generates the canonically associated tight Gabor frame minimizes among all windows generating a normalized tight Gabor frame. We present and prove versions of this result in the time domain, the frequency domain, the time-frequency domain, and the Zak transform domain, where in each domain the canonical is expressed using functional calculus for Gabor frame operators. Furthermore, we derive a Wiener-Levy type theorem for rationally oversampled Gabor frames. Finally, a Newton-type method for a fast numerical calculation of is presented. We analyze the convergence behavior of this method and demonstrate the efficiency of the proposed algorithm by some numerical examples.

Paper Structure

This paper contains 12 sections, 8 theorems, 150 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Representations of the Gabor frame operator
  3. Canonical tight Gabor frames and the Fan-Hoffman inequality
  4. The problem in the time domain and in the frequency domain
  5. The problem in the time domain
  6. The problem in the frequency domain
  7. The problem in the time-frequency domain
  8. The problem in the Zak transform domain
  9. A Wiener-Levy theorem for Gabor frames
  10. A Newton method for the computation of tight Gabor frames
  11. Convergence of the algorithm
  12. Numerical simulations

Key Result

Lemma 1.1

Let $T=(T_{kl})_{k,l \in {\mathbb Z}}$ be a hermitian positive definite biinfinite matrix with $c_1 I \le T \le c_2 I$ . Define $T_N = (T_{kl})_{|k|,|l| \le N}.$ Then $T_N^{-\frac{1}{2}} \rightarrow T^{-\frac{1}{2}}$ strongly for $N \rightarrow \infty$.

Figures (3)

  • Figure 1:
  • Figure 2: Comparison of different scalings for Newton's method.
  • Figure 3: Comparison of different methods to compute the tight window ${h^{0}}$.

Theorems & Definitions (13)

  • Lemma 1.1
  • proof
  • Theorem 2.1
  • Proposition 3.1
  • Theorem 3.2
  • Definition 3.3
  • Proposition 3.4
  • proof
  • Lemma 4.1
  • proof
  • ...and 3 more