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Basis beeldverwerking (8D040) dr. Andrea Fuster Prof.dr. Bart ter Haar Romeny dr. Anna Vilanova Prof.dr.ir. Marcel Breeuwer The Fourier Transform I

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Contents Complex numbers etc. Impulses Fourier Transform (+examples) Convolution theorem Fourier Transform of sampled functions Discrete Fourier Transform 2

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Introduction Jean Baptiste Joseph Fourier (* ) French Mathematician La Théorie Analitique de la Chaleur (1822) 3

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Fourier Series Any periodic function can be expressed as a sum of sines and/or cosines Fourier Series 4 (see figure 4.1 book)

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Fourier Transform Even functions that are not periodic and have a finite area under curve can 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 5

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Contents Complex numbers etc. Impulses Fourier Transform (+examples) Convolution theorem Fourier Transform of sampled functions Discrete Fourier Transform 6

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Complex numbers Complex number Its complex conjugate 7

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Complex numbers polar Complex number in polar coordinates 8

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Eulers formula 9 Sin (θ) Cos (θ) ? ?

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10 Re Im

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Complex math Complex (vector) addition Multiplication with i is rotation by 90 degrees in the complex plane 11

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Contents Complex number etc. Impulses Fourier Transform (+examples) Convolution theorem Fourier Transform of sampled functions Discrete Fourier Transform 12

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Unit impulse (Dirac delta function) Definition Constraint Sifting property Specifically for t=0 13

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Discrete unit impulse Definition Constraint Sifting property Specifically for x=0 14

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What does this look like? Impulse train 15 ΔT = 1 Note: impulses can be continuous or discrete!

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Contents Complex number etc. Impulses Fourier Transform (+examples) Convolution theorem Fourier Transform of sampled functions Discrete Fourier Transform 16

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Fourier Series with 17 Series of sines and cosines, see Eulers formula Periodic with period T

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Fourier transform – 1D cont. case 18 Symmetry: The only difference between the Fourier transform and its inverse is the sign of the exponential.

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Fourier Euler Fourier and Euler 19

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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 20

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Examples – Block W/2W/2 A

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Examples – Block 2 22

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Examples – Block 3 23 ?

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Examples – Impulse 24 constant

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Examples – Shifted impulse 25 Euler

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Examples – Shifted impulse 2 26 Real partImaginary part impulseconstant

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Also: using the following symmetry 27

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Examples - Impulse train 28 Periodic with period ΔT Encompasses only one impulse, so

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Examples - Impulse train 2 29

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Intermezzo: Symmetry in the FT 30

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So: the Fourier transform of an impulse train with period is also an impulse train with period 32

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Contents Complex number etc. Impulses Fourier Transform (+examples) Convolution theorem Fourier Transform of sampled functions Discrete Fourier Transform 33

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Fourier + Convolution What is the Fourier domain equivalent of convolution? 34

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What is 35

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Intermezzo 1 What is ? Let, so 36

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Intermezzo 2 Property of Fourier Transform 37

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Fourier + Convolution contd 38

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Convolution theorem Convolution in one domain is multiplication in the other domain: And also: 39

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And: Shift in one domain is multiplication with complex exponential (modulation) in the other domain And: 40

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Contents Complex number etc. Impulses Fourier Transform (+examples) Convolution theorem Fourier Transform of sampled functions Discrete Fourier Transform 41

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Sampling Idea: convert a continuous function into a sequence of discrete values. 42 (see figure 4.5 book)

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Sampling Sampled function can be written as Obtain value of arbitrary sample k as 43

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Sampling

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Sampling

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Sampling

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FT of sampled functions Fourier transform of sampled function Convolution theorem From FT of impulse train 47 (who?)

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FT of sampled functions 48

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Sifting property of is a periodic infinite sequence of copies of, with period 49

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Sampling Note that sampled function is discrete but its Fourier transform is continuous! 50

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Contents Complex number etc. Impulses Fourier Transform (+examples) Convolution theorem Fourier Transform of sampled functions Sampling theorem Aliasing Discrete Fourier Transform Application Examples 51

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Sampling theorem Band-limited function Sampled function lower value of 1/ΔT would cause triangles to merge 52

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Sampling theorem 2 Sampling theorem: If copy of can be isolated from the periodic sequence of copies contained in, can be completely recovered from the sampled version. Since is a continuous, periodic function with period 1/ΔT, one complete period from is enough to reconstruct the signal. This can be done via the Inverse FT. 53

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Extracting a single period from that is equal to is possible if Sampling theorem 3 54 Nyquist frequency

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Contents Complex number etc. Impulses Fourier Transform (+examples) Convolution theorem Fourier Transform of sampled functions Sampling theorem Aliasing Discrete Fourier Transform Application Examples 55

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Aliasing If, aliasing can occur 56

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Contents Complex number etc. Impulses Fourier Transform (+examples) Convolution theorem Fourier Transform of sampled functions Discrete Fourier Transform 57

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Discrete Fourier Transform Continuous transform of sampled function 58

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is continuous and infinitely periodic with period 1/ΔT 59

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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 60

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Substituting Into yields 61 Note: separation between samples in F. domain is

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By now we probably need some … 62

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