Shannon’s Sampling Theorem for Bandlimited Signals and Their Hilbert Transform, Boas-Type Formulae for Higher Order Derivatives—The Aliasing Error Involved by Their Extensions from Bandlimited to Non-Bandlimited Signals

doi:10.3390/e14112192

This article is

freely available
re-usable

Entropy 2012, 14(11), 2192-2226; doi:10.3390/e14112192

Article

Shannon’s Sampling Theorem for Bandlimited Signals and Their Hilbert Transform, Boas-Type Formulae for Higher Order Derivatives—The Aliasing Error Involved by Their Extensions from Bandlimited to Non-Bandlimited Signals^†

Paul L. Butzer¹, Gerhard Schmeisser² and Rudolf L. Stens^1,*

¹ Lehrstuhl A für Mathematik, RWTH Aachen University, 52056 Aachen, Germany ² Department of Mathematics, University of Erlangen-Nuernberg, 91058 Erlangen, Germany

^† Dedicated to Karl Willy Wagner (1883–1953). A man of principles, pioneer of the theory of electronic filters.

* Author to whom correspondence should be addressed.

Received: 29 August 2012; in revised form: 6 October 2012 / Accepted: 8 October 2012 / Published: 5 November 2012

(This article belongs to the Special Issue Information Theory Applied to Communications and Networking)

Download PDF Full-Text [396 KB, uploaded 5 November 2012 16:19 CET]

Abstract: The paper is concerned with Shannon sampling reconstruction formulae of derivatives of bandlimited signals as well as of derivatives of their Hilbert transform, and their application to Boas-type formulae for higher order derivatives. The essential aim is to extend these results to non-bandlimited signals. Basic is the fact that by these extensions aliasing error terms must now be added to the bandlimited reconstruction formulae. These errors will be estimated in terms of the distance functional just introduced by the authors for the extensions of basic relations valid for bandlimited functions to larger function spaces. This approach can be regarded as a mathematical foundation of aliasing error analysis of many applications.

Keywords: sampling formulae; differentiation formulae; non-bandlimited functions; aliasing error; Hilbert transforms; formulae with remainders; derivative-free error estimates; Bernstein’s inequality

Article Statistics

Click here to load and display the download statistics.

Cite This Article

MDPI and ACS Style

Butzer, P.L.; Schmeisser, G.; Stens, R.L. Shannon’s Sampling Theorem for Bandlimited Signals and Their Hilbert Transform, Boas-Type Formulae for Higher Order Derivatives—The Aliasing Error Involved by Their Extensions from Bandlimited to Non-Bandlimited Signals. Entropy 2012, 14, 2192-2226.

AMA Style

Butzer PL, Schmeisser G, Stens RL. Shannon’s Sampling Theorem for Bandlimited Signals and Their Hilbert Transform, Boas-Type Formulae for Higher Order Derivatives—The Aliasing Error Involved by Their Extensions from Bandlimited to Non-Bandlimited Signals. Entropy. 2012; 14(11):2192-2226.

Chicago/Turabian Style

Butzer, Paul L.; Schmeisser, Gerhard; Stens, Rudolf L. 2012. "Shannon’s Sampling Theorem for Bandlimited Signals and Their Hilbert Transform, Boas-Type Formulae for Higher Order Derivatives—The Aliasing Error Involved by Their Extensions from Bandlimited to Non-Bandlimited Signals." Entropy 14, no. 11: 2192-2226.

Entropy EISSN 1099-4300 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert

Shannon’s Sampling Theorem for Bandlimited Signals and Their Hilbert Transform, Boas-Type Formulae for Higher Order Derivatives—The Aliasing Error Involved by Their Extensions from Bandlimited to Non-Bandlimited Signals†

Article Statistics

Cite This Article

Shannon’s Sampling Theorem for Bandlimited Signals and Their Hilbert Transform, Boas-Type Formulae for Higher Order Derivatives—The Aliasing Error Involved by Their Extensions from Bandlimited to Non-Bandlimited Signals^†