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Transformación Discreta de Fourier

Module by: Phil Schniter. E-mail the authorTranslated By: Fara Meza, Erika Jackson

Based on: Discrete Fourier Transformation by Phil Schniter

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

N-puinto Punto Transformada Discreta de Fourier (DFT)

Xk=n=0N1xne(j)2πnkn   ,   k=0N1    X k n 0 N 1 x n 2 n k n k k 0 N 1
(1)
xn=1Nk=0N1Xkej2πnkn   ,   n=0N1    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=0N1    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 :
    Xej2πMk= n =0N1xne(j)2πMk X 2 M k n 0 N 1 x n 2 M k
    (3)
    Xej2πMk=n=0N1 x zp ne(j)2πMk X 2 M k n 0 N 1 x zp n 2 M k Xej2πMk= X zp k   ,   k=0M1    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:
    Xejω= n =0N1xne(j)ωn X ω n 0 N 1 x n ω n
    (4)
    Xejω=n=0N11Nk=0N1Xkej2πNkne(j)ωn X ω n 0 N 1 1 N k 0 N 1 X k 2 N k n ω n Xejω=k=0N1Xk1Nk=0N1e(j)(ω2πNk)n X ω k 0 N 1 X k 1 N k 0 N 1 ω 2 N k n Xejω=k=0N1Xk1N(sinωN2πk2sinωN2πk2Ne(j)(ω2πNk)N12) 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(j)2πN W N 2 N ,
    ( X0 X1 XN1 )=( 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 )( x0 x1 xN1 ) 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: (1NW)1NWH=1NWH(1NW)=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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