Signal Processing Course : Fourier
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This document discusses Fourier processing and covers topics such as continuous and discrete Fourier bases, sampling, 2D Fourier bases, and Fourier approximation. The continuous Fourier basis uses complex exponentials while the discrete Fourier basis uses a finite number of complex exponentials. Sampling and periodization allow transforming between continuous and discrete settings. The 2D Fourier basis is a product of 1D bases. Fourier approximation represents functions as a sum of complex exponentials.
![Fourier and Convolution
f ∗1[ − 1 , 1 ]
2 2
f
x− 1
2
x x+ 1
2](https://image.slidesharecdn.com/course-signal-fourier-121213052749-phpapp02/85/Signal-Processing-Course-Fourier-6-320.jpg)
![Fourier and Convolution
f ∗1[ − 1 , 1 ]
2 2
f
x− 1
2
x x+ 1
2
1
0.8
0.6
0.4
0.2
0
1
0.8
0.6
0.4
0.2
0
1
0.8
0.6
0.4
0.2
0](https://image.slidesharecdn.com/course-signal-fourier-121213052749-phpapp02/85/Signal-Processing-Course-Fourier-7-320.jpg)
![Fourier and Convolution
f ∗1[ − 1 , 1 ]
2 2
f
x− 1
2
x x+ 1
2
1
0.8
0.6
0.4
0.2
0
1
0.8
0.6
0.4
0.2
0
1
0.8
0.6
0.4
0.2
0](https://image.slidesharecdn.com/course-signal-fourier-121213052749-phpapp02/85/Signal-Processing-Course-Fourier-8-320.jpg)
![Discrete Fourier Transform
ˆ ˆ
g [m] = f [m] · h[m]
ˆ](https://image.slidesharecdn.com/course-signal-fourier-121213052749-phpapp02/85/Signal-Processing-Course-Fourier-12-320.jpg)
![Discrete Fourier Transform
ˆ ˆ
g [m] = f [m] · h[m]
ˆ](https://image.slidesharecdn.com/course-signal-fourier-121213052749-phpapp02/85/Signal-Processing-Course-Fourier-13-320.jpg)
![The Four Settings
Note: for Fourier, bounded periodic.
Infinite continuous domains: f0 (t), t R ... ...
+⇥
ˆ
f0 ( ) = f0 (t)e i t
dt
⇥
Periodic continuous domains: f0 (t), t ⇥ [0, 1] R/Z
1
ˆ
f0 [m] = f0 (t)e 2i mt
dt
0
Infinite discrete domains: f [n], n Z ... ...
ˆ
f( ) = f [n]ei n
n Z
Periodic discrete domains: f [n], n ⇤ {0, . . . , N 1} ⇥ Z/N Z
N 1
ˆ
f [m] = f [n]e
2i
N mn
n=0](https://image.slidesharecdn.com/course-signal-fourier-121213052749-phpapp02/85/Signal-Processing-Course-Fourier-15-320.jpg)
![Fourier Transforms
Periodization f0 (t) ⇥ n f0 (t + n)
f [n] = f0 (n/N )
Sampling
f0 (t), t R f0 (t), t [0, 1] Continuous
f [n], n Z f [n], 0 n<N Discrete
Fourier transform
Isometry f ⇥ f ˆ
Infinite Periodic
f0 (N (⇥ + 2k ))
ˆ
Sampling f0 ( ) ⇥ ˆ
{f0 (k)}k
ˆ
f0 ( ), ˆ
f0 [k], k Z Infinite
Periodization
R
ˆ ˆ
ˆ
f (⇥), ⇥ [0, 2 ] f [k], 0 k<N Periodic
k
f (⇥) =
Continuous Discrete
ˆ
+⇥ 1
ˆ
f0 ( ) = f0 (t)e i t
dt ˆ
f0 [m] = f0 (t)e 2i mt
dt
⇥ 0
N 1
ˆ
f( ) = f [n]e i n ˆ
f [m] = f [n]e
2i
N mn
n Z n=0](https://image.slidesharecdn.com/course-signal-fourier-121213052749-phpapp02/85/Signal-Processing-Course-Fourier-16-320.jpg)
![2D Fourier Basis
1 2i m1 n1 + 2i m2 n2
em [n] = e N0 N0
= em1 [n1 ]em2 [n2 ]
N](https://image.slidesharecdn.com/course-signal-fourier-121213052749-phpapp02/85/Signal-Processing-Course-Fourier-24-320.jpg)
![2D Fourier Basis
1 2i m1 n1 + 2i m2 n2
em [n] = e N0 N0
= em1 [n1 ]em2 [n2 ]
N](https://image.slidesharecdn.com/course-signal-fourier-121213052749-phpapp02/85/Signal-Processing-Course-Fourier-25-320.jpg)
Signal Processing Course : Fourier
- 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. Continuous Fourier Bases Continuous Fourier basis: m (x) = em (x) = e2i mx
- 4. Continuous Fourier Bases Continuous Fourier basis: m (x) = em (x) = e2i mx
- 6. Fourier and Convolution f ∗1[ − 1 , 1 ] 2 2 f x− 1 2 x x+ 1 2
- 7. Fourier and Convolution f ∗1[ − 1 , 1 ] 2 2 f x− 1 2 x x+ 1 2 1 0.8 0.6 0.4 0.2 0 1 0.8 0.6 0.4 0.2 0 1 0.8 0.6 0.4 0.2 0
- 8. Fourier and Convolution f ∗1[ − 1 , 1 ] 2 2 f x− 1 2 x x+ 1 2 1 0.8 0.6 0.4 0.2 0 1 0.8 0.6 0.4 0.2 0 1 0.8 0.6 0.4 0.2 0
- 9. Overview • Continuous Fourier Basis • Discrete Fourier Basis • Sampling • 2D Fourier Basis • Fourier Approximation
- 12. Discrete Fourier Transform ˆ ˆ g [m] = f [m] · h[m] ˆ
- 13. Discrete Fourier Transform ˆ ˆ g [m] = f [m] · h[m] ˆ
- 14. Overview • Continuous Fourier Basis • Discrete Fourier Basis • Sampling • 2D Fourier Basis • Fourier Approximation
- 15. The Four Settings Note: for Fourier, bounded periodic. Infinite continuous domains: f0 (t), t R ... ... +⇥ ˆ f0 ( ) = f0 (t)e i t dt ⇥ Periodic continuous domains: f0 (t), t ⇥ [0, 1] R/Z 1 ˆ f0 [m] = f0 (t)e 2i mt dt 0 Infinite discrete domains: f [n], n Z ... ... ˆ f( ) = f [n]ei n n Z Periodic discrete domains: f [n], n ⇤ {0, . . . , N 1} ⇥ Z/N Z N 1 ˆ f [m] = f [n]e 2i N mn n=0
- 16. Fourier Transforms Periodization f0 (t) ⇥ n f0 (t + n) f [n] = f0 (n/N ) Sampling f0 (t), t R f0 (t), t [0, 1] Continuous f [n], n Z f [n], 0 n<N Discrete Fourier transform Isometry f ⇥ f ˆ Infinite Periodic f0 (N (⇥ + 2k )) ˆ Sampling f0 ( ) ⇥ ˆ {f0 (k)}k ˆ f0 ( ), ˆ f0 [k], k Z Infinite Periodization R ˆ ˆ ˆ f (⇥), ⇥ [0, 2 ] f [k], 0 k<N Periodic k f (⇥) = Continuous Discrete ˆ +⇥ 1 ˆ f0 ( ) = f0 (t)e i t dt ˆ f0 [m] = f0 (t)e 2i mt dt ⇥ 0 N 1 ˆ f( ) = f [n]e i n ˆ f [m] = f [n]e 2i N mn n Z n=0
- 17. Sampling and Periodization (a) (b) 1 (c) 0 (d)
- 18. Sampling and Periodization: Aliasing (a) (b) 1 (c) 0 (d)
- 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 1 2i m1 n1 + 2i m2 n2 em [n] = e N0 N0 = em1 [n1 ]em2 [n2 ] N
- 25. 2D Fourier Basis 1 2i m1 n1 + 2i m2 n2 em [n] = e N0 N0 = em1 [n1 ]em2 [n2 ] N
- 26. Overview • Continuous Fourier Basis • Discrete Fourier Basis • Sampling • 2D Fourier Basis • Fourier Approximation
- 28. 1D Fourier Approximation 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0