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Revista de Matemática Teoría y Aplicaciones

Print version ISSN 1409-2433

Rev. Mat vol.28 n.2 San José Jul./Dec. 2021

http://dx.doi.org/10.15517/rmta.v28i2.43356 

Artículo

Discrete sampling theorem to Shannon’s sampling theorem using the hyperreal numbers ∗R

Del teorema del muestreo discreto a teorema del muestreo de Shannon mediante los números hiperreales ∗R

José L. Simancas-García1 

Kemel George-González2 

1Universidad de la Costa, Departamento de Ciencias de la Computación y Electrónica, Barranquilla, Colombia; jsimanca3@cuc.edu.co

2CINVESTAV, México, & Fundación Innovación y Conocimiento, Barranquilla, Colombia; kemel.george@gmail.com

Abstract

Shannon’s sampling theorem is one of the most important results of modern signal theory. It describes the reconstruction of any band-limited signal from a finite number of its samples. On the other hand, although less well known, there is the discrete sampling theorem, proved by Cooley while he was working on the development of an algorithm to speed up the calculations of the discrete Fourier transform. Cooley showed that a sampled signal can be resampled by selecting a smaller number of samples, which reduces computational cost. Then it is possible to reconstruct the original sampled signal using a reverse process. In principle, the two theorems are not related. However, in this paper we will show that in the context of Non-Standard Mathematical Analysis (NSA) and Hyperreal Numerical System ∗R, the two theorems are equivalent. The difference between them becomes a matter of scale. With the scale changes that the hyperreal number system allows, the discrete variables and functions become continuous, and Shannon’s sampling theorem emerges from the discrete sampling theorem.

Keywords: Sampling theorem; subsampling; hyperreal number system; infinitesimal calculus model.

Resumen

El teorema del muestreo de Shannon es uno de los resultados más importantes de la moderna teoría de señales. Este describe la reconstrucción de toda señal de banda limitada desde un número finito de sus muestras. Por otra parte, aunque menos conocido, se tiene el teorema del muestreo discreto, demostrado por Cooley mientras trabajaba en la elaboración de un algoritmo para acelerar los cálculos de la transformada discreta de Fourier. Cooley demostró que una señal muestreada se puede volver a muestrearla mediante la selección de un número menor de muestras, lo cual reduce el costo computacional. Luego, es posible reconstruir la señal muestreada original mediante un proceso inverso. En principio, los dos teoremas no están relacionados. Sin embargo, en este artíclo demostraremos que, en el contexto del Análisis Matemático No Estándar (ANS) y el Sistema Numérico Hiperreal ∗R, los dos teoremas son equivalentes. La diferencia entre ellos se vuelve un asunto de escala. Con los cambios de escala que permite realizar el sistema numérico hiperreal, las variables y funciones discretas se vuelven continuas, y el teorema del muestreo de Shannon emerge del teorema del muestreo discreto.

Palabras clave: Teorema de Muestreo; Submuestreo; Sistema Numérico Hiperreal; Modelo de Cálculo Infinitesimal.

Mathematics Subject Classification: 94D02

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Acknowledgements

We thank Universidad de la Costa for giving us the opportunity to present these non-standard constructions in the courses Advanced Mathematics for Engineering, Signals and Systems and Digital Signal Processing. We also want to thank the students in these courses for the patience and interest in learning non-standard ways of studying engineering.

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Received: October 16, 2020; Revised: April 15, 2021; Accepted: May 19, 2021

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