Signal Processing Course: Fourier Processing

Transcript

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