Skip to content Skip to navigation

Connexions

You are here: Home » Content » Discrete Fourier Transformation

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

Lenses

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.

This content is ...

In these lenses

  • richb's DSP

    This module is included inLens: richb's DSP resources
    By: Richard BaraniukAs a part of collection:"Signals and Systems"

    Comments:

    "My introduction to signal processing course at Rice University."

    Click the "richb's DSP" link to see all content selected in this lens.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.

Discrete Fourier Transformation

Module by: Phil Schniter

Summary: This module covers the fundamentals of Discrete-Fourier Transformations.

N-point Discrete Fourier Transform (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 that:

  • Xk X k is the DTFT evaluated at ω=2πNk ∀k,k=0…N-1 ω 2 N k k k 0 … N 1
  • Zero-padding xn x n to MM samples prior to the DFT yields an MM-point uniform sampled version of the 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
  • The NN-pt DFT is sufficient to reconstruct the entire DTFT of an NN-pt sequence:
    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

Figure 1: Dirichlet sinc, 1NsinωN2sinω2 1 N ω N 2 ω 2
Figure 1 (dirichletsinc.png)

  • The DFT has a convenient matrix representation. Defining 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)
    where X=Wx X W x respectively. WW has the following properties:
    • WW is Vandermonde: the nnth column of WW is a polynomial in W N n W N n
    • WW is symmetric: W=WT W W
    • 1NW 1 N W is unitary: 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 , the IDFT matrix.
  • For NN a power of 2, the FFT can be used to compute the DFT using about N2log2N N 2 2 N rather than N2 N 2 operations.

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