Open AccessDiscrete sampling theorem to Shannon’s sampling theorem using the hyperreal numbers R
Author(s) -
José L. Simancas-García,
Kemel George-González
Publication year - 2021
Publication title -
revista de matemáticas
Language(s) - English
Resource type - Journals
eISSN - 2215-3373
pISSN - 1409-2433
DOI - 10.15517/rmta.v28i2.43356
Subject(s) - nyquist–shannon sampling theorem , sampling (signal processing) , mathematics , signal (programming language) , fourier transform , scale (ratio) , fundamental theorem , discrete fourier transform (general) , fourier analysis , discrete mathematics , algorithm , mathematical analysis , computer science , telecommunications , fixed point theorem , fractional fourier transform , physics , quantum mechanics , detector , programming language
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.