Gabriel Peyré
January 01, 2012
1.5k

# Signal Processing Course: Fourier Processing

January 01, 2012

## Transcript

1. Fourier
Processing
Gabriel Peyré

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