Skip to content Skip to navigation

Connexions

You are here: Home » Content » DFT as a Matrix Operation

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.

      What are tags? tag icon

      Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

    • External bookmarks
  • E-mail the author
  • Rate this module (How does the rating system work?)

    Rating system

    Ratings

    Ratings allow you to judge the quality of modules. If other users have ranked the module then its average rating is displayed below. Ratings are calculated on a scale from one star (Poor) to five stars (Excellent).

    How to rate a module

    Hover over the star that corresponds to the rating you wish to assign. Click on the star to add your rating. Your rating should be based on the quality of the content. You must have an account and be logged in to rate content.

    		(0 ratings)

Recently Viewed

This feature requires Javascript to be enabled.

DFT as a Matrix Operation

Module by: Robert Nowak

Summary: This module introduces linear algebra, DFT, FFT, matrix and vector.

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

Matrix Review

Recall:

  • Vectors in ℝN N : ∀xi,xi∈ℝ:x=x0x1…x N - 1 xi xi x x0 x1 … x N - 1
  • Vectors in ℂN N : ∀xi,xi∈ℂ:x=x0x1…x N - 1 xi xi x x0 x1 … x N - 1
  • Transposition:
    1. transpose: xT=x0x1…x N - 1 x x0 x1 … x N - 1
    2. conjugate: xH=x0¯x1¯…x N - 1 ¯ x x0 x1 … x N - 1
  • Inner product:
    1. real: xTy=∑i=0N−1xiyi x y i 0 N 1 xi yi
    2. complex: xHy=∑i=0N−1xn¯yn x y i 0 N 1 xn yn
  • Matrix Multiplication: Ax= a 0 0 a 0 1 …a 0 , N - 1 a 1 0 a 1 1 …a 1 , N - 1 ⋮⋮…⋮a N - 1 , 0 a N - 1 , 1 …a N - 1 , N - 1 x0x1…x N - 1 =y0y1…y N - 1 A x a 0 0 a 0 1 … a 0 , N - 1 a 1 0 a 1 1 … a 1 , N - 1 ⋮ ⋮ … ⋮ a N - 1 , 0 a N - 1 , 1 … a N - 1 , N - 1 x0 x1 … x N - 1 y0 y1 … y N - 1 yk=∑n=0N−1a k ⁢ n xn yk n 0 N 1 a k ⁢ n xn
  • Matrix Transposition: AT= a 0 0 a 1 0 …a N - 1 , 0 a 0 1 a 1 1 …a N - 1 , 1 ⋮⋮…⋮a 0 , N - 1 a 1 , N - 1 …a N - 1 , N - 1 A a 0 0 a 1 0 … a N - 1 , 0 a 0 1 a 1 1 … a N - 1 , 1 ⋮ ⋮ … ⋮ a 0 , N - 1 a 1 , N - 1 … a N - 1 , N - 1 Matrix transposition involved simply swapping the rows with columns. AH=AT¯ A A The above equation is Hermitian transpose. ATkn=Ank A k n A n k AHkn=A¯nk A k n A n k

Representing DFT as Matrix Operation

Now let's represent the DFT in vector-matrix notation. x=x0x1…xN−1 x x 0 x 1 … x N 1 X=X0X1…XN−1∈ℂN X X 0 X 1 … X N 1 N Here xx is the vector of time samples and XX is the vector of DFT coefficients. How are xx and XX related: Xk=∑n=0N−1xnⅇ-ⅈ2πNkn X k n 0 N 1 x n 2 N k n where a k ⁢ n =ⅇ-ⅈ2πNkn=WNkn a k ⁢ n 2 N k n WN k n so X=Wx X W x where XX is the DFT vector, WW is the matrix and xx the time domain vector. Wkn=ⅇ-ⅈ2πNkn W k n 2 N k n X=Wx0x1…xN−1 X W x 0 x 1 … x N 1 IDFT: xn=1N∑k=0N−1Xkⅇⅈ2πNnk x n 1 N k 0 N 1 X k 2 N n k where ⅇⅈ2πNnk=WNnk¯ 2 N n k WN n k WNnk¯ WN n k is the matrix Hermitian transpose. So, x=1NWHX x 1 N W X where xx is the time vector, 1NWH 1 N W is the inverse DFT matrix, and XX is the DFT vector.

Comments, questions, feedback, criticisms?

Send feedback