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Transformación Discreta de Fourier

Module by: Phil Schniter Translated By Fara Meza, Erika JacksonBased on: Discrete Fourier Transformation by Phil Schniter

Summary: Este modulo cubre los fundamentos de las Transformada Discreta de Fourier.

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N-puinto Punto Transformada Discreta de Fourier (DFT)

Xk=∑n=0N−1xnⅇ-ⅈ2πnkn ∀k,k=0…N−1 X k n 0 N 1 x n 2 n k n k k 0 … N 1 (1)
xn=1N∑k=0N−1Xkⅇⅈ2πnkn ∀n,n=0…N−1 x n 1 N k 0 N 1 X k 2 n k n n n 0 … N 1 (2)

Note que:

  • Xk X k es la DTFT evaluado en ω=2πNk ∀k,k=0…N−1 ω 2 N k k k 0 … N 1
  • Completar con ceros xn x n a MM muestras antes de sacar el DFT, da como resultado una versión muestreada de MM-puntos uniformes del DTFT :
    Xⅇⅈ2πMk=∑n=0N−1xnⅇ-ⅈ2πMk X 2 M k n 0 N 1 x n 2 M k (3)
    Xⅇⅈ2πMk=∑n=0N−1 x zp nⅇ-ⅈ2πMk X 2 M k n 0 N 1 x zp n 2 M k Xⅇⅈ2πMk= X zp k ∀k,k=0…M−1 X 2 M k X zp k k k 0 … M 1
  • La NN-pt DFT es suficiente para reconstruir toda la DTFT de una secuencia de NN-pt:
    Xⅇⅈω=∑n=0N−1xnⅇ-ⅈωn X ω n 0 N 1 x n ω n (4)
    Xⅇⅈω=∑n=0N−11N∑k=0N−1Xkⅇⅈ2πNknⅇ-ⅈωn X ω n 0 N 1 1 N k 0 N 1 X k 2 N k n ω n Xⅇⅈω=∑k=0N−1Xk1N∑k=0N−1ⅇ-ⅈω−2πNkn X ω k 0 N 1 X k 1 N k 0 N 1 ω 2 N k n Xⅇⅈω=∑k=0N−1Xk1NsinωN−2πk2sinωN−2πk2Nⅇ-ⅈω−2πNkN−12 X ω k 0 N 1 X k 1 N ω N 2 k 2 ω N 2 k 2 N ω 2 N k N 1 2

Figura 1: Sinc dirichlet, 1NsinωN2sinω2 1 N ω N 2 ω 2
Figura 1 (dirichletsinc.png)

  • DFT tiene una representación en forma de matriz muy conveniente. Definiendo W N =ⅇ-ⅈ2πN W N 2 N ,
    X0X1⋮XN−1= W N 0 W N 0 W N 0 W N 0 … W N 0 W N 1 W N 2 W N 3 … W N 0 W N 2 W N 4 W N 6 …⋮⋮⋮⋮⋮x0x1⋮xN−1 X 0 X 1 ⋮ X N 1 W N 0 W N 0 W N 0 W N 0 … W N 0 W N 1 W N 2 W N 3 … W N 0 W N 2 W N 4 W N 6 … ⋮ ⋮ ⋮ ⋮ ⋮ x 0 x 1 ⋮ x N 1 (5)
    donde X=Wx X W x respectivamente. WW tiene las siguientes propiedades:
    • WW es Vandermonde: La nnth columna de WW es un polinomio en W N n W N n
    • WW es simetrico: W=WT W W
    • 1NW 1 N W es unitaria: 1NW1NWH=1NWH1NW=I 1 N W 1 N W H 1 N W H 1 N W I
    • 1NW¯=W-1 1 N W W -1 , es la matriz y DFT.
  • • Para NN un poder de 2, la FFT se puede usar para calcular la DFT usando N2log2N N 2 2 N en vez de N2 N 2 operaciones.

Tabla 1
N N N2log2N N 2 2 N N2 N 2
16 32 256
64 192 4096
256 1024 65536
1024 5120 1048576

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