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Este modulo describe la Transformada rápida de Fourier (FFT).

IntroducciÓN

La Transformada Rápida de Fourier (Fast Fourier Transform) (FFT) es un algoritmo eficiente O(NlogN) para calcular la DFT

  • orignalmente descubierta por Gauss a primcipios de 1800
  • redescubierta por Cooley y Tukey en IBM durante 1960
  • C.S. Burrus, de la Universidad de Rice University siendo jefe del departamento de Ingenieria, literalmente "escribio el libro" de los algoritmos de la rápida Transformada Discreta de Fourier DFT.
La FFT explota las simetrias en la matriz W para aproximarse "divide y conquistaras". No hablaremos del actual algoritmo de la FFT aqui, veamos estas notas si usted esta interesado en leer mas a cerca de la idea detras de la FFT.

ComparaciÓN de la velocidad

¿En cuánto es mejor O(NlogN) que O( N 2 )?

Esta figura muestra que tan lento crece el tiempo de solución de un proceso de O(NlogN).
N 10 100 1000 10 6 10 9
N 2 100 10 4 10 6 10 12 10 18
N N 1 200 3000 6 10 6 9 10 9

Digamos que tiene una maquina de 1 MFLOP (un millión de "puntos flotantes" de operacione spor segundo). Sea N 1 millión 10 6 .

Un algoritmo de O( N 2 ) toma 10 12 procesos→ 10 6 segundos≃11.5 días.

Un algoritmo de O( N N ) toma 6 10 6 procesos→6 segundos.

N 1 millión es razonable.

Una camara digital de 3 megapixeles arroja 3 10 6 números por cada foto. Asíque para dos N secuencias de punto f n y h n . Si resolvemos directamente f n h n : O( N 2 ) operaciones.

Tomando la FFTs -- O(NlogN)

multiplicando la FFTs -- O(N)

la inversa de FFTs -- O(NlogN).

el total de complejidad es O(NlogN).

FFT + computación didigital fue completamente responsable de la "explosión" del Procesamiento Digital de Señales DSP en los años 60's.

Got questions? Get instant answers now!

La Universidad de Rice fue (y sigue siendo) uno de los lugares de hacer investigación en DSP.

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Source:  OpenStax, Señales y sistemas. OpenStax CNX. Sep 28, 2006 Download for free at http://cnx.org/content/col10373/1.2
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