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11: Transformadas Discretas de Fourier

Última actualización

30 oct 2022
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- 10.6: Integrando ODEs con Scipy
- 11.1: Conversión de Transformada Continua de Fourier a DFT

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La Transformada Discreta de Fourier (DFT) es una versión discretizada de la transformada de Fourier, la cual es ampliamente utilizada en simulación y análisis numéricos. Dado un conjunto de $N$ números $\{f_0, f_1, \dots, f_{N-1}\}$ , el DFT produce otro conjunto de $N$ números $N$ números $\{F_0, F_1, \dots, F_{N-1}\}$ , definidos de la siguiente manera:

$\mathrm{DFT}\Big\{f_0, f_1, \dots, f_{N-1}\Big\} = \Big\{F_0, F_1, \dots, F_{N-1}\Big\} \qquad\mathrm{where}\quad F_n = \sum_{m=0}^{N-1} e^{-2\pi i \frac{mn}{N}}\, f_m.$

La inversa de esta transformación es la Transformada Discreta Inversa de Fourier (IDFT):

$\mathrm{IDFT}\Big\{F_0, F_1, \dots, F_{N-1}\Big\} = \Big\{f_0, f_1, \dots, f_{N-1}\Big\} \qquad\mathrm{where}\quad f_m = \frac{1}{N} \sum_{n=0}^{N-1} e^{2\pi i \frac{mn}{N}}\, F_n.$

La relación inversa entre la DFT y la IDFT es sencilla de probar, mediante el uso de la identidad

$\sum_{m=0}^{N-1} e^{\pm 2\pi i \frac{m(n-n')}{N}} = N \delta_{nn'},$

donde $\delta_{nn'}$ denota el delta de Kronecker. Esta identidad se deriva de la fórmula geométrica de la serie.

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