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Sinh regularized Lagrangian nonuniform sampling series

Haixin Jiang, Xinyu Chen, Liang Chen

TL;DR

Addresses exponential convergence in nonuniform sampling of bandlimited signals by introducing a sinh-type regularization into the Lagrangian nonuniform sampling framework. It develops the theory for Paley-Wiener spaces $\mathcal{B}_{\delta}(\mathbb{R})$ and sine-type interpolation, defining sinh-regularized nonuniform series $S_{f,Q,N,\varphi}$ and periodic variant $S_{f,\psi,N,\varphi}$ using $\varphi_{\beta,m}$ and $\varphi_{\beta M,(N-1)M}$. It proves that sinh-regularized schemes achieve nearly twice the convergence rate of Gaussian-regularized counterparts and corroborates this with numerical experiments. The results enhance reconstruction accuracy in nonuniform sampling and guide window design for fast convergence in practical signal processing.

Abstract

Recently, some window functions have been introduced into the nonuniform fast Fourier transform and the regularized Shannon sampling. Inspired by these works, we utilize a sinh-type function to accelerate the convergence of the Lagrangian nonuniform sampling series. Our theoretical error estimates and numerical experiments demonstrate that the sinh regularized nonuniform sampling series achieves a superior convergence rate compared to the fastest existing Gaussian regularized nonuniform sampling series.

Sinh regularized Lagrangian nonuniform sampling series

TL;DR

Addresses exponential convergence in nonuniform sampling of bandlimited signals by introducing a sinh-type regularization into the Lagrangian nonuniform sampling framework. It develops the theory for Paley-Wiener spaces and sine-type interpolation, defining sinh-regularized nonuniform series and periodic variant using and . It proves that sinh-regularized schemes achieve nearly twice the convergence rate of Gaussian-regularized counterparts and corroborates this with numerical experiments. The results enhance reconstruction accuracy in nonuniform sampling and guide window design for fast convergence in practical signal processing.

Abstract

Recently, some window functions have been introduced into the nonuniform fast Fourier transform and the regularized Shannon sampling. Inspired by these works, we utilize a sinh-type function to accelerate the convergence of the Lagrangian nonuniform sampling series. Our theoretical error estimates and numerical experiments demonstrate that the sinh regularized nonuniform sampling series achieves a superior convergence rate compared to the fastest existing Gaussian regularized nonuniform sampling series.
Paper Structure (3 sections, 62 equations, 2 figures, 3 tables)

This paper contains 3 sections, 62 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Sinh regularized Lagrangian nonuniform sampling series
  3. Numerical experiments

Figures (2)

  • Figure 3.1: Logarithm of reconstruction errors by the sinh-type regularized non-periodic sampling formula (\ref{['sinhformula']}). "bound"$=\log_{10}(\exp(-(N-1)(\pi-\delta))$.
  • Figure 3.2: Logarithm of reconstruction errors by the sinh-type regularized periodic sampling formula (\ref{['periodicsinhformula']}). "bound"$=\log_{10}(\exp(-(N-1)M(\pi-\delta))$.