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

Matrix review

Recall:

  • Vectors in N : x i x i x x 0 x 1 x N - 1
  • Vectors in N : x i x i x x 0 x 1 x N - 1
  • Transposition:
    • transpose: x x 0 x 1 x N - 1
    • conjugate: x x 0 x 1 x N - 1
  • Inner product :
    • real: x y i 0 N 1 x i y i
    • complex: x y i 0 N 1 x n y n
  • Matrix Multiplication: 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 x 0 x 1 x N - 1 y 0 y 1 y N - 1 y k n 0 N 1 a k n x n
  • Matrix Transposition: 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. A A The above equation is Hermitian transpose. A k 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 x 0 x 1 x N 1 X X 0 X 1 X N 1 N Here x is the vector of time samples and X is the vector of DFT coefficients. How are x and X related: X k n 0 N 1 x n 2 N k n where a k n 2 N k n W N k n so X W x where X is the DFT vector, W is the matrix and x the time domain vector. W k n 2 N k n X W x 0 x 1 x N 1 IDFT: x n 1 N k 0 N 1 X k 2 N n k where 2 N n k W N n k W N n k is the matrix Hermitian transpose. So, x 1 N W X where x is the time vector, 1 N W is the inverse DFT matrix, and X is the DFT vector.

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Source:  OpenStax, Digital signal processing: a user's guide. OpenStax CNX. Aug 29, 2006 Download for free at http://cnx.org/content/col10372/1.2
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