Signal Processing Course : Fourier

Fourier

Processing

Gabriel Peyréhttp://www.ceremade.dauphine.fr/~peyre/numerical-tour/

Overview

•Continuous Fourier Basis

•Discrete Fourier Basis

•Sampling

•2D Fourier Basis

•Fourier Approximation

�m(x) = em(x) = e2i�mx

Continuous Fourier Bases

Continuous Fourier basis:

�m(x) = em(x) = e2i�mx

Continuous Fourier Bases

Continuous Fourier basis:

Fourier and Convolution

Fourier and Convolution

x! x+x 12

12

f

f

"1[! 12 ,

12 ]

Fourier and Convolution

x! x+x 12

12

f

f

"1[! 12 ,

12 ]

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Fourier and Convolution

x! x+x 12

12

f

f

"1[! 12 ,

12 ]

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Overview

•Continuous Fourier Basis

•Discrete Fourier Basis

•Sampling

•2D Fourier Basis

•Fourier Approximation

Discrete Fourier Transform

Discrete Fourier Transform

Discrete Fourier Transform

g[m] = f [m] · h[m]

Discrete Fourier Transform

g[m] = f [m] · h[m]

Overview

•Continuous Fourier Basis

•Discrete Fourier Basis

•Sampling

•2D Fourier Basis

•Fourier Approximation

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

f [m] =N�1�

n=0

f [n]e�2i�N mn

f0(�) =� +⇥

�⇥f0(t)e�i�tdt

f0[m] =� 1

0f0(t)e�2i�mtdt

f(�) =�

n�Zf [n]ei�n

Note: for Fourier, bounded � periodic.

. . . . . .

. . .. . .f [n], n � Z

f [n], n ⇤ {0, . . . , N � 1} ⇥ Z/NZ

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

Continuousf0(t), t � R f0(t), t � [0, 1]

f0(t) ⇥��

n f0(t + n)

Sam

plin

g

f0(�) ⇥� {f0(k)}k

Discrete

Infinite

Periodic

Continuous

Sampling

f [k], 0 � k < N

f0(�),� � R f0[k], k � Z Four

ier

tran

sfor

mIs

omet

ryf⇥�

f

f0(�) =� +⇥

�⇥f0(t)e�i�tdt f0[m] =

� 1

0f0(t)e�2i�mtdt

f(�) =�

n�Zf [n]ei�n

f(⇥),⇥ � [0, 2�]

Peri

odiz

atio

nf(⇥

)=

� k

f 0(N

(⇥+

2k�))

f[n

]=f 0

(n/N

)

Sampling and Periodization

(a)

(c)

(d)

(b)

1

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Sampling and Periodization: Aliasing

(b)

(c)

(d)

(a)

0

1

Uniform Sampling and Smoothness

Uniform Sampling and Smoothness

Uniform Sampling and Smoothness

Uniform Sampling and Smoothness

Overview

•Continuous Fourier Basis

•Discrete Fourier Basis

•Sampling

•2D Fourier Basis

•Fourier Approximation

2D Fourier Basis

em[n] =1�N

e2i�N0

m1n1+ 2i�N0

m2n2 = em1 [n1]em2 [n2]

2D Fourier Basis

em[n] =1�N

e2i�N0

m1n1+ 2i�N0

m2n2 = em1 [n1]em2 [n2]

Overview

•Continuous Fourier Basis

•Discrete Fourier Basis

•Sampling

•2D Fourier Basis

•Fourier Approximation

1D Fourier Approximation

1D Fourier Approximation

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2D Fourier Approximation