Basis beeldverwerking (8D040)

dr. Andrea Fusterdr. Anna VilanovaProf.dr.ir. Marcel Breeuwer

The Fourier Transform I

Contents

• Complex numbers etc.• Impulses• Fourier Transform (+examples)• Convolution theorem• Fourier Transform of sampled functions• Discrete Fourier Transform

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Introduction• Jean Baptiste

Joseph Fourier (*1768-†1830)

• French Mathematician• La Théorie Analitique

de la Chaleur (1822)

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Fourier Series

• Any periodic function can be expressed as a sum of sines and/or cosines

Fourier Series

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(see figure 4.1 book)

Fourier Transform

• Even functions that • are not periodic • and have a finite area under curvecan be expressed as an integral of sines and cosines multiplied by a weighing function

• Both the Fourier Series and the Fourier Transform have an inverse operation:

• Original Domain Fourier Domain

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Contents

• Complex numbers etc.• Impulses• Fourier Transform (+examples)• Convolution theorem• Fourier Transform of sampled functions• Discrete Fourier Transform

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Complex numbers

• Complex number

• Its complex conjugate

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Complex numbers polar

• Complex number in polar coordinates

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Euler’s formula

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Sin (θ)

Cos (θ)

?

?

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Re

Im

Complex math

• Complex (vector) addition

• Multiplication with i

is rotation by 90 degrees in the complex plane

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Contents

• Complex number etc.• Impulses• Fourier Transform (+examples)• Convolution theorem• Fourier Transform of sampled functions• Discrete Fourier Transform

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Unit impulse (Dirac delta function)

• Definition

• Constraint

• Sifting property

• Specifically for t=0

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Discrete unit impulse

• Definition

• Constraint

• Sifting property

• Specifically for x=0

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What does this look like?

Impulse train

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ΔT = 1

Note: impulses can be continuous or discrete!

Contents

• Complex number etc.• Impulses• Fourier Transform (+examples)• Convolution theorem• Fourier Transform of sampled functions• Discrete Fourier Transform

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Fourier Series

with

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Series of sines and cosines,

see Euler’s formula

Periodic with

period T

Fourier transform – 1D cont. case

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Symmetry: The only difference between the Fourier transform and its inverse is the sign of the exponential.

Fourier

Euler

Fourier and Euler

• If f(t) is real, then F(μ) is complex• F(μ) is expansion of f(t) multiplied by sinusoidal terms• t is integrated over, disappears• F(μ) is a function of only μ, which determines the

frequency of sinusoidals• Fourier transform frequency domain

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Examples – Block 1

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-W/2 W/2

A

Examples – Block 2

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Examples – Block 3

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?

Examples – Impulse

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constant

Examples – Shifted impulse

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Euler

Examples – Shifted impulse 2

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Real part Imaginary part

impulse constant

• Also: using the following symmetry

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Examples - Impulse train

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Periodic with period ΔT

Encompasses only one impulse, so

Examples - Impulse train 2

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• So: the Fourier transform of an impulse train with period is also an impulse train with period

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Contents

• Complex number etc.• Impulses• Fourier Transform (+examples)• Convolution theorem• Fourier Transform of sampled functions• Discrete Fourier Transform

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Fourier + Convolution

• What is the Fourier domain equivalent of convolution?

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• What is

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Intermezzo 1

• What is ?

• Let , so

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Intermezzo 2

• Property of Fourier Transform

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Fourier + Convolution cont’d

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Convolution theorem

• Convolution in one domain is multiplication in the other domain:

• And also:

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And:

• Shift in one domain is multiplication with complex exponential (modulation) in the other domain

• And:

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Contents

• Complex number etc.• Impulses• Fourier Transform (+examples)• Convolution theorem• Fourier Transform of sampled functions• Discrete Fourier Transform

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Sampling

• Idea: convert a continuous function into a sequence of discrete values.

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(see figure 4.5 book)

Sampling

• Sampled function can be written as

• Obtain value of arbitrary sample k as

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Sampling - 2

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Sampling - 3

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FT of sampled functions

• Fourier transform of sampled function

• Convolution theorem

• From FT of impulse train

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(who?)

FT of sampled functions

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• Sifting property

• of is a periodic infinite sequence of copies of , with period

49(see figure 4.6 in Gonzalez & Woods)

Sampling

• Note that sampled function is discrete but its Fourier transform is continuous!

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Contents

• Complex number etc.• Impulses• Fourier Transform (+examples)• Convolution theorem• Fourier Transform of sampled functions• Discrete Fourier Transform

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Discrete Fourier Transform

• Idea: given a sampled function, obtain its sampled Fourier transform

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Discrete Fourier Transform

• Continuous transform of sampled function

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• is continuous and infinitely periodic with period 1/ΔT

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• We need only one period to characterize• If we want to take M equally spaced samples from

in the period μ = 0 to μ = 1/Δ, this can be done thus

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• Substituting

• Into

• yields

62Note: separation between samples in F. domain is

(see example 4.4 book)

By now we probably need some …

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