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DFT as a Matrix Operation

Module by: Robert Nowak

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

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.

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