Signal Processing Course: Fourier Processing

E34ded36efe4b7abb12510d4e525fee8?s=47 Gabriel Peyré
January 01, 2012

Signal Processing Course: Fourier Processing

E34ded36efe4b7abb12510d4e525fee8?s=128

Gabriel Peyré

January 01, 2012
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  1. Fourier Processing Gabriel Peyré http://www.ceremade.dauphine.fr/~peyre/numerical-tour/

  2. Overview •Continuous Fourier Basis •Discrete Fourier Basis •Sampling •2D Fourier

    Basis •Fourier Approximation
  3. m (x) = em (x) = e2i mx Continuous Fourier

    Bases Continuous Fourier basis:
  4. m (x) = em (x) = e2i mx Continuous Fourier

    Bases Continuous Fourier basis:
  5. Fourier and Convolution

  6. Fourier and Convolution x− x+ x 1 2 1 2

    f f ∗1[− 1 2 , 1 2 ]
  7. Fourier and Convolution x− x+ x 1 2 1 2

    f f ∗1[− 1 2 , 1 2 ] 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
  8. Fourier and Convolution x− x+ x 1 2 1 2

    f f ∗1[− 1 2 , 1 2 ] 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
  9. Overview •Continuous Fourier Basis •Discrete Fourier Basis •Sampling •2D Fourier

    Basis •Fourier Approximation
  10. Discrete Fourier Transform

  11. Discrete Fourier Transform

  12. Discrete Fourier Transform ˆ g[m] = ˆ f[m] · ˆ

    h[m]
  13. Discrete Fourier Transform ˆ g[m] = ˆ f[m] · ˆ

    h[m]
  14. Overview •Continuous Fourier Basis •Discrete Fourier Basis •Sampling •2D Fourier

    Basis •Fourier Approximation
  15. Infinite continuous domains: Periodic continuous domains: Infinite discrete domains: Periodic

    discrete domains: f0 (t), t R f0 (t), t ⇥ [0, 1] R/Z The Four Settings ˆ f[m] = N 1 n=0 f[n]e 2i N mn ˆ f0 ( ) = +⇥ ⇥ f0 (t)e i tdt ˆ f0 [m] = 1 0 f0 (t)e 2i mtdt ˆ f( ) = n Z f[n]ei n Note: for Fourier, bounded periodic. .. . .. . .. . .. . f[n], n Z f[n], n ⇤ {0, . . . , N 1} ⇥ Z/NZ
  16. ˆ f[m] = N 1 n=0 f[n]e 2i N mn

    Fourier Transforms Discrete Infinite Periodic f[n], n Z f[n], 0 n < N Periodization Continuous f0 (t), t R f0 (t), t [0, 1] f0 (t) ⇥ n f0 (t + n) Sampling ˆ f0 ( ) ⇥ { ˆ f0 (k)}k Discrete Infinite Periodic Continuous Sampling ˆ f[k], 0 k < N ˆ f0 ( ), R ˆ f0 [k], k Z Fourier transform Isometry f ⇥ ˆ f ˆ f0 ( ) = +⇥ ⇥ f0 (t)e i tdt ˆ f0 [m] = 1 0 f0 (t)e 2i mtdt ˆ f( ) = n Z f[n]ei n ˆ f(⇥), ⇥ [0, 2 ] Periodization ˆ f(⇥) = k ˆ f0 (N(⇥ + 2k )) f[n] = f0 (n/N)
  17. Sampling and Periodization (a) (c) (d) (b) 1 0

  18. Sampling and Periodization: Aliasing (b) (c) (d) (a) 0 1

  19. Uniform Sampling and Smoothness

  20. Uniform Sampling and Smoothness

  21. Uniform Sampling and Smoothness

  22. Uniform Sampling and Smoothness

  23. Overview •Continuous Fourier Basis •Discrete Fourier Basis •Sampling •2D Fourier

    Basis •Fourier Approximation
  24. 2D Fourier Basis em [n] = 1 N e 2i

    N0 m1n1+ 2i N0 m2n2 = em1 [n1 ]em2 [n2 ]
  25. 2D Fourier Basis em [n] = 1 N e 2i

    N0 m1n1+ 2i N0 m2n2 = em1 [n1 ]em2 [n2 ]
  26. Overview •Continuous Fourier Basis •Discrete Fourier Basis •Sampling •2D Fourier

    Basis •Fourier Approximation
  27. 1D Fourier Approximation

  28. 1D Fourier Approximation 0 0.2 0.4 0.6 0.8 1 0

    0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
  29. 2D Fourier Approximation