Fourier Transforms

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# Fourier Transforms - PowerPoint PPT Presentation

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

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

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 series
• The coefficients become

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

Fourier series
• Alternate forms
• where

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

Complex exponential notation
• Euler’s formula

Phasor notation:

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

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
• 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 form
• Thus
• So you could also write

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

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 transform

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

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 transform
• Ordinary frequency

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

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 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

Theorem

Proof (1)

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