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Properties of Fourier Transforms

  1. Linearity
     
    If ab C, then
     
    Description: 4077.png
     
    provided the Fourier transforms of f(t) and g(t) exist.
  2. Scaling
     
    If F{f (t)} = F(ω) and c  R, then
     
    Description: 4083.png
  3. Time shifting
     
    If Description: 4089.png, then
     
    Description: 4095.png
  4. Frequency shifting
     
    If Description: 4102.png, then
     
    Description: 4108.png
  5. Symmetry
     
    If Description: 4114.png, then
     
    Description: 4120.png
  6. Modulation
     
    If Description: 4126.png, then
     
    Description: 4132.png
     
    Description: 4138.png
  7. Differentiation in time
     
    Let n  N and suppose that f (n) is piecewise continuous. Assume that Description: 4152.png, then
     
    Description: 4158.png
     
    In particular
     
    Description: 4167.png
     
    and
     
    Description: 4173.png
  8. Frequency differentiation
     
    Let n  N and suppose that f is piecewise continuous. Then
     
    Description: 4179.png
     
    In particular
     
    Description: 4186.png
     
    and
     
    Description: 4192.png




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