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DFT as a Matrix Operation

Module by: Robert Nowak. E-mail the author

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

Matrix Review

Recall:

  • Vectors in RN N : x=( x0 x1 x N - 1 )  ,   xi R    xi xi x x0 x1 x N - 1
  • Vectors in CN N : x=( x0 x1 x N - 1 )  ,   xi C    xi xi x x0 x1 x N - 1
  • Transposition:
    1. transpose: xT=( x0 x1 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=0N1 xi yi x y i 0 N 1 xi yi
    2. complex: xHy=i=0N1 xn * 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 )( x0 x1 x N - 1 )=( y0 y1 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=0N1 a 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. ATk,n=An,k A k n A n k AHk,n=A*n,k A k n A n k

Representing DFT as Matrix Operation

Now let's represent the DFT in vector-matrix notation. x=( x0 x1 xN1 ) x x 0 x 1 x N 1 X=( X0 X1 XN1 )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=0N1xne(j2πNkn) X k n 0 N 1 x n 2 N k n where a k n =e(j2πN)kn= WN kn 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. Wk,n=e(j2πN)kn W k n 2 N k n X=Wx0x1xN1 X W x 0 x 1 x N 1 IDFT: xn=1Nk=0N1Xkej2πNnk x n 1 N k 0 N 1 X k 2 N n k where ej2πNnk= WN nk* 2 N n k WN n k WN nk* 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.

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