## Articulo

• Similares en SciELO

## versión impresa ISSN 1409-2433

### Rev. Mat vol.28 no.2 San José jul./dic. 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

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

Ver contenido completo en PDF.

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.

References

O,E,Brigham.The Fast Fourier Transform and Its Applications, Prentice Hall, Englewood Cliffs NJ, 1988. http://sar.kangwon.ac.kr/gisg/FFTbook.pdf Links ]

W, Cooley; P,A,W, Lewis; P,D, Welch. The Finite Fourier Transform, IEEE Transactions on Audio and Electroacoustics 12(1969), no. 1, 27-34. Doi: 10.1109/TE.1969.4320436 [ Links ]

H,V,Dannon. Infinitesimals, Gauge Institute Journal 6(2010), no. 4, 1-34. http://www.gauge-institute.org/Infinitesimal/infinitesimals.pdf Links ]

H,V,Dannon. Infinitesimals Calculus, Gauge Institute Journal 7(2011), no. 4, 1-57. http://www.gauge-institute.org/Infinitesimal/infinitesimalCalculus.pdf Links ]

K, George-González. El cálculo discreto infinitesimal y la didáctica de la transformada de Fourier. Ph.D. Thesis, Centro de Investigación y de Estudios Avanzados del IPN, Ciudad de México, 1998. [ Links ]

K, George-González. Cálculo con Infinitesimales, Editorial Universidad del Magdalena, Santa Marta, Colombia, 2001. [ Links ]

K, George-González. Conversión del dominio discreto en dominio continuo: Otro enfoque de aprendizaje, Revista Escenarios 15(2017), no. 1, 160-168. Doi: 10.15665/esc.v15i1.1172 [ Links ]

S, Haykin. Communication Systems, John Wiley and Sons, New York NY, 2001. [ Links ]

J,M, Henle; E,M, Kleinberg. Infinitesimal Calculus, Dover Publications, New York NY, 1979. [ Links ]

B,B,Hubbard.The World According to Wavelets. The Story of a Mathematical Technique in the Making, A.K. Peters, Natick MA, 1998. [ Links ]

A,J,Jerri.The Shannon sampling theorem-its various extensions and applications: A tutorial review, Proceedings of the IEEE 65(1977), no. 11, 1565-1596. Doi: 10.1109/PROC.1977.10771 [ Links ]

J,M,López. Cálculo de infinitésimos, Departamento de Matemáticas - Universidad de Puerto Rico, Puerto Rico, 2014. [ Links ]

W,A,J,Luxemburg. Infinitesimal Calculus, Dover Publications, New York NY, 1962. [ Links ]

E, Margolis; Y,C, Eldar. Nonuniform Sampling of Periodic Bandlimited Signals, IEEE Transactions on Signal Processing 56(2008), no. 7, 2728-2745. Doi: 10.1109/TSP.2008.917416 [ Links ]

G, Meinsma; L, Mirkin. Sampling from a system theoretic viewpoint: Part I-Concepts and tools, IEEE Transactions on Signal Processing 58(2010), no. 7, 3578-3590. Doi: 10.1109/TSP.2010.2047641 [ Links ]

G, Meinsma; L, Mirkin. Sampling from a system theoretic viewpoint: Part II-Noncausal solutions, IEEE Transactions on Signal Processing 58(2010), no. 7, 3591-3606. Doi: 10.1109/TSP.2010.2047642 [ Links ]

H, Nyquist. Certain topics in telegraph transmission theory, Transactions of the American Institute of Electrical Engineers, 47(1928), no. 2, 617-644. Doi: 10.1109/T-AIEE.1928.5055024 [ Links ]

J,G, Proakis; D,G, Manolakis. Digital Signal Processing: Principles, Algorithms and Applications, Prentice Hall, New York, 1995. https://engineering.purdue.edu/ ee538/DSPT ext3rdEdition.pdfLinks ]

A, Robinson. Non-Standard Analysis, Princeton University Press, Princeton NJ, 1996. [ Links ]

C,E, ShannonCommunication in the presence of noise, Proceedings of the Institute of Radio Engeneers 37(1949), no. 1, 10-21. Doi: 10.1109/JRPROC.1949.232969 [ Links ]

J,L, Simancas-García; K, George-González. Signals and Linear Systems: A Novel Approach Based on Infinitesimal Calculus (Part I), IEEE Latin America Transactions 18(2020), no. 11, 1953-1965. Doi: 10.1109/TLA.2020.9398637 [ Links ]

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