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Quantitative Assessment of Dual window Efficiency in Gabor Frame Expansions

Sruthi Raghoothaman, Noufal Asharaf

TL;DR

This work investigates reconstruction quality in Gabor frame expansions using compactly supported windows $B_2$, $B_3$, and $\varepsilon_3$ by comparing canonical and multiple alternative dual windows. It applies two dual-construction strategies—a perturbation-based method and a recursive, explicit scheme—across three window families and evaluates reconstruction accuracy with the Average Mean Squared Error on five standard signals. The results show that exponential B-splines $\varepsilon_3$ consistently yield the lowest AMSE, and among duals, symmetric constructions like $k$ (and their perturbations) deliver the most accurate reconstructions, offering practical guidance for time-frequency analysis tasks. Together, these findings provide concrete guidance on selecting window types and duals to achieve stable, high-fidelity Gabor reconstructions in signal processing applications.

Abstract

We investigate the performance of Gabor frame reconstructions using three compactly supported window functions: the second-order B-spline ($B_2$), the third-order B-spline ($B_3$), and the exponential B-spline of order 3 ($\varepsilon_3$). For each generator, various dual windows are considered, including the canonical dual, symmetric and asymmetric duals, and perturbation-based duals constructed via a recent duality result. The reconstruction quality is assessed using the Average Mean Squared Error (AMSE) across five standard test signals: Blocks, Bumps, Heavisine, Doppler, and Quadchirp. Numerical experiments demonstrate that exponential B-splines yield the lowest AMSE among the three, confirming their effectiveness in Gabor frame applications. Among the dual windows, the symmetric dual and its perturbation-based variant consistently achieve the best reconstruction accuracy, making them strong candidates for practical use in signal processing tasks.

Quantitative Assessment of Dual window Efficiency in Gabor Frame Expansions

TL;DR

This work investigates reconstruction quality in Gabor frame expansions using compactly supported windows , , and by comparing canonical and multiple alternative dual windows. It applies two dual-construction strategies—a perturbation-based method and a recursive, explicit scheme—across three window families and evaluates reconstruction accuracy with the Average Mean Squared Error on five standard signals. The results show that exponential B-splines consistently yield the lowest AMSE, and among duals, symmetric constructions like (and their perturbations) deliver the most accurate reconstructions, offering practical guidance for time-frequency analysis tasks. Together, these findings provide concrete guidance on selecting window types and duals to achieve stable, high-fidelity Gabor reconstructions in signal processing applications.

Abstract

We investigate the performance of Gabor frame reconstructions using three compactly supported window functions: the second-order B-spline (), the third-order B-spline (), and the exponential B-spline of order 3 (). For each generator, various dual windows are considered, including the canonical dual, symmetric and asymmetric duals, and perturbation-based duals constructed via a recent duality result. The reconstruction quality is assessed using the Average Mean Squared Error (AMSE) across five standard test signals: Blocks, Bumps, Heavisine, Doppler, and Quadchirp. Numerical experiments demonstrate that exponential B-splines yield the lowest AMSE among the three, confirming their effectiveness in Gabor frame applications. Among the dual windows, the symmetric dual and its perturbation-based variant consistently achieve the best reconstruction accuracy, making them strong candidates for practical use in signal processing tasks.

Paper Structure

This paper contains 7 sections, 6 theorems, 48 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Construction of Dual Windows
  4. Numerical Results
  5. B Spline
  6. Exponential B spline
  7. AMSE Analysis of Gabor Frame Reconstructions

Key Result

Theorem 2.1

12 Let $a,b>0$ be given. Suppose that $g \in L^2(\mathbb{R})$ has support in an interval of length $\frac{1}{b}$. Suppose there exist $A,B>0$ such that Then $\mathcal{G}(g,a,b)$ is a frame for $L^2(\mathbb{R})$ and the canonical dual generator is given by

Figures (4)

  • Figure 4.1: Sampled test signals used in the experiments, each sampled at 2048 points in the interval $[0,1]$.
  • Figure 4.2: $a = 1$, $b = \frac{1}{5}$. Green curve represents the generator $B_2$. (a) Blue: symmetric dual $k$; red: the dual constructed from $k$ using the Stoeva method. (b) Blue: asymmetric dual $h$; red: the dual constructed from $h$ using Proposition \ref{['gdual']}. (c) Blue: canonical dual $S^{-1}B_2$; red: the dual constructed from it using Proposition \ref{['gdual']}.
  • Figure 4.3: $a = 1$, $b = \frac{1}{5}$. Green curve represents the generator $B_3$. (a) Blue: symmetric dual $k$; red: the dual constructed from $k$ using the Stoeva method. (b) Blue: asymmetric dual $h$; red: the dual constructed from $h$ using Proposition \ref{['gdual']}. (c) Blue: canonical dual $S^{-1}B_3$; red: the dual constructed from it using Proposition \ref{['gdual']}.
  • Figure 4.4: $a = 1$, $b = \frac{1}{5}$. Green curve represents the generator $\varepsilon_3$. (a) Blue: symmetric dual $k$; red: the dual constructed from $k$ using the Stoeva method. (b) Blue: asymmetric dual $h$; red: the dual constructed from $h$ using Proposition \ref{['gdual']}. (c) Blue: canonical dual $S^{-1}\varepsilon_3$; red: the dual constructed from it using Proposition \ref{['gdual']}.

Theorems & Definitions (8)

  • Theorem 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Proposition 3.1
  • Proposition 3.2
  • Theorem 3.1