Skip to content Skip to navigation

Connexions

You are here: Home » Content » Transformación Discreta de Fourier

Navigation

Content Actions
  • Download module PDF
  • Add to ...
    Add the module to:
    • My Favorites
    • A lens
    • An external social bookmarking service
    • My Favorites (What is 'My Favorites'?)
      'My Favorites' is a special kind of lens which you can use to bookmark modules and collections directly in Connexions. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need a Connexions account to use 'My Favorites'.
    • A lens (What is a lens?)

      Definition of a lens

      Lenses

      A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

      What is in a lens?

      Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

      Who can create a lens?

      Any individual Connexions member, a community, or a respected organization.

    • External bookmarks
  • E-mail the author

Recently Viewed

This feature requires Javascript to be enabled.

Transformación Discreta de Fourier

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

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

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.

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

Comments, questions, feedback, criticisms?

Send feedback