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La Transformada Rápida de Fourier (FFT)

Module by: Justin Romberg. E-mail the authorTranslated By: Fara Meza, Erika Jackson

Based on: The Fast Fourier Transform (FFT) by Justin Romberg

Summary: 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 WW 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( N2 N 2 )?

Figura 1: Esta figura muestra que tan lento crece el tiempo de solución de un proceso de O(NlogN).
Figura 1 (fft_f1s.png)
Tabla 1
NN 1010 100100 10001000 106106 109109
N2N2 100 104104 106106 10121012 10181018
N⁢logNNN 11 200200 30003000 6×1066106 9×1099109

Digamos que tiene una maquina de 1 MFLOP (un millión de "puntos flotantes" de operacione spor segundo). Sea N=1⁢millión=106 N 1 millión 10 6 .

Un algoritmo de O( N2 N 2 ) toma 10121012 procesos → 106 10 6 segundos ≃ 11.5 días.

Un algoritmo de O( N⁢logN N N ) toma 6×106 6 10 6 procesos → 6 segundos.

nota:

N=1⁢millión N 1 millión es razonable.

Ejemplo 1

Una camara digital de 3 megapixeles arroja 3×106 3 10 6 números por cada foto. Así que para dos NN secuencias de punto f⁢nfn y h⁢nhn. Si resolvemos directamente f⁢n⊛h⁢n ⊛ f n h n : O( N2 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).

nota:

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

nota:

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

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