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Wigner measures in the discrete setting: high-frequency analysis of sampling & reconstruction operators

Fabricio Macia

TL;DR

This work develops a rigorous microlocal (Wigner) framework for understanding how sampling and reconstruction on regular grids modify high-frequency oscillations and concentrations in $L^{2}$-bounded sequences. It introduces discrete Wigner measures $M^{\varepsilon}[U^{h}]$ and derives explicit formulas connecting the Wigner measures of sampled and reconstructed sequences to the original continuous measures, notably $\mu_{\varphi,\psi}(x,\xi)=|\widehat{\psi}(\xi)|^{2}\sum_{n\in\mathbb{Z}^{d}}|\widehat{\varphi}(\xi+2\pi n)|^{2}\mu(x,\xi+2\pi n)$ and $\mu_{\varphi}(x,\xi)=|\widehat{\varphi}(\xi)|^{2}\mu(x,\xi)$. The paper also links defect measures to Wigner measures, analyzes regimes $h\sim\varepsilon$ and $h\ll\varepsilon$, establishes Poisson-summation-based proofs, and provides counterexamples showing limits of $h$-oscillation preservation. Together these results yield criteria for when sampling/reconstruction preserve, filter, or distort high-frequency content, with Shannon-like special cases illustrating band-limited reconstruction. The framework enables precise quantification of oscillation-direction effects and informs design of profiles $(\varphi,\psi)$ to achieve desired filtering or preservation properties in numerical sampling schemes.

Abstract

The goal of this article is that of understanding how the oscillation and concentration effects developed by a sequence of functions in $\mathbb{R}^{d} $ are modified by the action of Sampling and Reconstruction operators on regular grids. Our analysis is performed in terms of Wigner and defect measures, which provide a quantitative description of the high frequency behavior of bounded sequences in $L^{2}(mathbb{R}^{d}) $. We actually present explicit formulas that make possible to compute such measures for sampled/reconstructed sequences. As a consequence, we are able to characterize sampling and reconstruction operators that preserve or filter the high-frequency behavior of specific classes of sequences. The proofs of our results rely on the construction and manipulation of Wigner measures associated to sequences of discrete functions.

Wigner measures in the discrete setting: high-frequency analysis of sampling & reconstruction operators

TL;DR

This work develops a rigorous microlocal (Wigner) framework for understanding how sampling and reconstruction on regular grids modify high-frequency oscillations and concentrations in -bounded sequences. It introduces discrete Wigner measures and derives explicit formulas connecting the Wigner measures of sampled and reconstructed sequences to the original continuous measures, notably and . The paper also links defect measures to Wigner measures, analyzes regimes and , establishes Poisson-summation-based proofs, and provides counterexamples showing limits of -oscillation preservation. Together these results yield criteria for when sampling/reconstruction preserve, filter, or distort high-frequency content, with Shannon-like special cases illustrating band-limited reconstruction. The framework enables precise quantification of oscillation-direction effects and informs design of profiles to achieve desired filtering or preservation properties in numerical sampling schemes.

Abstract

The goal of this article is that of understanding how the oscillation and concentration effects developed by a sequence of functions in are modified by the action of Sampling and Reconstruction operators on regular grids. Our analysis is performed in terms of Wigner and defect measures, which provide a quantitative description of the high frequency behavior of bounded sequences in . We actually present explicit formulas that make possible to compute such measures for sampled/reconstructed sequences. As a consequence, we are able to characterize sampling and reconstruction operators that preserve or filter the high-frequency behavior of specific classes of sequences. The proofs of our results rely on the construction and manipulation of Wigner measures associated to sequences of discrete functions.

Paper Structure

This paper contains 28 sections, 43 theorems, 207 equations.

Table of Contents

  1. Introduction
  2. Statement of the problem: oscillation and concentration under the effect of sampling and reconstruction
  3. Wigner measures
  4. Computation of Wigner and defect measures
  5. Strategy of proof: Wigner measures in the discrete setting
  6. Plan of the article
  7. Notations and conventions
  8. Sampling and reconstruction
  9. Definitions and examples
  10. Boundedness properties
  11. Bases and projections
  12. High frequency analysis: $h\sim \varepsilon$
  13. Reduction to the case $h=\varepsilon$
  14. Sampling
  15. Reconstruction
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $\left( u_{k}\right)$ be a bounded sequence in $L^{2}\left( \mathbb{R}^{d}\right)$ and let $\left( \varepsilon _{k}\right)$ be a sequence of positive numbers tending to zero. Then it is possible to extract a subsequence $\left( u_{k_{n}}\right)$ such that, for every test function $a\in \mathcal{ where $\mu$ is a finite positive measure on $\mathbb{R}^{d}\times \mathbb{R}^{d}$.

Theorems & Definitions (59)

  • Theorem 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 1.7
  • Proposition 1.8
  • Remark 1.9
  • Definition 2.1
  • ...and 49 more