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Fourier Transforms. Fourier series. To go from f(  ) to f( t ) substitute To deal with the first basis vector being of length 2 instead of , rewrite as. Fourier series. The coefficients become. Fourier series. Alternate forms where. Complex exponential notation. Euler’s formula.

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Fourier Transforms


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fourier transforms

Fourier Transforms

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell

fourier series
Fourier series
  • To go from f( ) to f(t) substitute
  • To deal with the first basis vector being of length 2 instead of , rewrite as

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell

fourier series3
Fourier series
  • The coefficients become

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell

fourier series4
Fourier series
  • Alternate forms
    • where

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell

complex exponential notation
Complex exponential notation
  • Euler’s formula

Phasor notation:

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell

euler s formula
Euler’s formula
  • Taylor series expansions
  • Even function ( f(x) = f(-x) )
  • Odd function ( f(x) = -f(-x) )

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell

complex exponential form
Complex exponential form
  • Consider the expression
  • So
  • Since an and bnare real, we can let

and get

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell

complex exponential form8
Complex exponential form
  • Thus
  • So you could also write

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell

fourier transform
Fourier transform
  • We now have
  • Let’s not use just discrete frequencies, n0, we’ll allow them to vary continuously too
  • We’ll get there by setting t0=-T/2 and taking limits as T and n approach 

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell

fourier transform10
Fourier transform

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell

fourier transform11
Fourier transform
  • So we have (unitary form, angular frequency)
  • Alternatives (Laplace form, angular frequency)

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell

fourier transform12
Fourier transform
  • Ordinary frequency

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell

fourier transform13
Fourier transform
  • Some sufficient conditions for application
    • Dirichlet conditions
      • f(t) has finite maxima and minima within any finite interval
      • f(t) has finite number of discontinuities within any finite interval
    • Square integrable functions (L2 space)
    • Tempered distributions, like Dirac delta

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell

fourier transform14
Fourier transform
  • Complex form – orthonormal basis functions for space of tempered distributions

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell

convolution theorem
Convolution theorem

Theorem

Proof (1)

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell