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On numerical realizations of Shannon's sampling theorem

Melanie Kircheis, Daniel Potts, Manfred Tasche

TL;DR

The paper addresses the poor numerical realization of Shannon's sampling theorem, particularly the slow convergence and noise sensitivity of the classical Shannon series. It systematically compares two regularization paradigms—frequency-window and time-window windowing—under oversampling, deriving algebraic decay for frequency-based methods and exponential decay for time-based methods. The key contributions are rigorous error bounds showing exponential decay for sinh-type and continuous Kaiser–Bessel time windows, and explicit results highlighting localized, interpolating sampling with strong numerical robustness. The findings guide practical reconstruction of bandlimited signals from finite, noisy samples, favoring time-domain regularization with appropriate window choices for efficient and stable implementations.

Abstract

In this paper, we discuss some numerical realizations of Shannon's sampling theorem. First we show the poor convergence of classical Shannon sampling sums by presenting sharp upper and lower bounds of the norm of the Shannon sampling operator. In addition, it is known that in the presence of noise in the samples of a bandlimited function, the convergence of Shannon sampling series may even break down completely. To overcome these drawbacks, one can use oversampling and regularization with a convenient window function. Such a window function can be chosen either in frequency domain or in time domain. We especially put emphasis on the comparison of these two approaches in terms of error decay rates. It turns out that the best numerical results are obtained by oversampling and regularization in time domain using a sinh-type window function or a continuous Kaiser-Bessel window function, which results in an interpolating approximation with localized sampling. Several numerical experiments illustrate the theoretical results.

On numerical realizations of Shannon's sampling theorem

TL;DR

The paper addresses the poor numerical realization of Shannon's sampling theorem, particularly the slow convergence and noise sensitivity of the classical Shannon series. It systematically compares two regularization paradigms—frequency-window and time-window windowing—under oversampling, deriving algebraic decay for frequency-based methods and exponential decay for time-based methods. The key contributions are rigorous error bounds showing exponential decay for sinh-type and continuous Kaiser–Bessel time windows, and explicit results highlighting localized, interpolating sampling with strong numerical robustness. The findings guide practical reconstruction of bandlimited signals from finite, noisy samples, favoring time-domain regularization with appropriate window choices for efficient and stable implementations.

Abstract

In this paper, we discuss some numerical realizations of Shannon's sampling theorem. First we show the poor convergence of classical Shannon sampling sums by presenting sharp upper and lower bounds of the norm of the Shannon sampling operator. In addition, it is known that in the presence of noise in the samples of a bandlimited function, the convergence of Shannon sampling series may even break down completely. To overcome these drawbacks, one can use oversampling and regularization with a convenient window function. Such a window function can be chosen either in frequency domain or in time domain. We especially put emphasis on the comparison of these two approaches in terms of error decay rates. It turns out that the best numerical results are obtained by oversampling and regularization in time domain using a sinh-type window function or a continuous Kaiser-Bessel window function, which results in an interpolating approximation with localized sampling. Several numerical experiments illustrate the theoretical results.
Paper Structure (5 sections, 13 theorems, 145 equations, 7 figures, 1 table)

This paper contains 5 sections, 13 theorems, 145 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Poor convergence of Shannon sampling sums
  3. Regularization with a frequency window function
  4. Regularization with a time window function
  5. Comparison of the two regularization methods

Key Result

Lemma 2.1

The linear operator $S_T \colon \,C_0(\mathbb R) \to C_0(\mathbb R)$ has the norm

Figures (7)

  • Figure 2.1: The norm $\|\tilde{f} - f \|_{C_0(\mathbb R)}$ of \ref{['eq:ftilde-f']} as well as its lower/upper bounds \ref{['eq:lower_bound']} and \ref{['eq:upper_bound']} for several $T \in \mathbb N$ and $L=N(1+\lambda)$ with $\lambda \in \{0,0.5,1,2\}$, where $N=128$ and $\varepsilon=10^{-3}$ are chosen.
  • Figure 3.1: The frequency window function \ref{['eq:hatpsilin(v)']} and its scaled inverse Fourier transform \ref{['eq:psilin']}.
  • Figure 3.2: Maximum approximation error \ref{['eq:err_psilin']} (solid) and error constant \ref{['eq:max|f-PTf|']} (dashed) using the linear frequency window $\psi_{\mathrm{lin}}$ from \ref{['eq:psilin']} in \ref{['eq:PTf']} for the function $f(t) = \sqrt{N} \,\mathrm{sinc}(N \pi t)$ with $N=128$, $T = 2^c$, $c \in \{0,\dots,15\}$, and $\lambda\in\{0.5,1,2\}$.
  • Figure 3.3: The frequency window functions \ref{['eq:hatpsi3(v)']} and \ref{['eq:hatpsi_cos(v)']}, and their scaled inverse Fourier transforms.
  • Figure 4.1: The regularized $\mathrm{sinc}$ function $\textcolor{black}{ \xi}_{\mathrm{cKB}}(t) \coloneqq \mathrm{sinc}(L\pi t) \, \varphi_{\mathrm{cKB}}(t)$ using the continuous Kaiser--Bessel window function \ref{['eq:varphicKB']} and its Fourier transform $\textcolor{black}{ \hat{\xi}}_{\mathrm{cKB}}$.
  • ...and 2 more figures

Theorems & Definitions (23)

  • Lemma 2.1
  • Theorem 2.2
  • Remark 2.3
  • Theorem 2.4
  • Remark 2.5
  • Example 3.1
  • Remark 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Example 3.5
  • ...and 13 more