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Discrete-Time Fourier Transform

Module by: Phil Schniter

Summary: Review of the discrete time fourier transform.

Z-Transform

Using xn x n to denote a discrete-time signal at index n∈ℤ n ℤ ,

Z-Transform

Xz=∑n=-∞∞xnz-n Xz n x n z n (1)

Discrete-Time Fourier Transform

Discrete-Time Fourier Transform

Xⅇⅈω=∑n=-∞∞xnⅇ-ⅈωn ∀ω,ω∈ℝ X ω n x n ω n ω ω (2)
xn=12π∫-∞∞Xⅇⅈωⅇⅈωndω ∀n,n∈ℤ x n 1 2 ω X ω ω n n n (3)
Note that:
  • Xⅇⅈω X ω is the z-transfer evaluated on the unit circle in ℂ-plane: z=ⅇⅈ z ω .
  • Xⅇⅈω X ω is 2π 2 -periodic in ωω.
  • xn∈ℝ x n implies Xⅇⅈω=Xⅈ-ω¯ X ω X ω , i.e., conjugate symmetry
Other DTFT properties are:
  • x-n↔Xⅇ-ⅈω ↔ x n X ω
  • xn¯↔Xⅇ-ⅈω¯ ↔ x n X ω
  • xn-ℓ↔Xⅇⅈωⅇ-ⅈωℓ ↔ x n ℓ X ω ω ℓ
  • xnⅇⅈ ω 0 n↔Xⅇⅈω- ω 0 ↔ x n ω 0 n X ω ω 0
  • xnyn↔12π∫-ππXⅇⅈYⅇⅈω-θdθ ↔ x n y n 1 2 θ X θ Y ω θ
  • xnyn↔XⅇⅈωYⅇⅈω ↔ x n y n X ω Y ω

Linear Convolution

xnyn=∑m=-∞∞xmyn-m x n y n m x m y n m (4)

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