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The Fourier Transform I

Basis beeldverwerking (8D040) dr. Andrea Fuster Prof.dr . Bart ter Haar Romeny dr. Anna Vilanova Prof.dr.ir . Marcel Breeuwer. The Fourier Transform I. Contents. Complex numbers etc. Impulses Fourier Transform (+examples) Convolution theorem Fourier Transform of sampled functions

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The Fourier Transform I

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  1. Basis beeldverwerking (8D040)dr. Andrea FusterProf.dr. Bart terHaarRomenydr. Anna VilanovaProf.dr.ir. Marcel Breeuwer The Fourier Transform I

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

  3. Introduction Jean BaptisteJoseph Fourier(*1768-†1830) French Mathematician La ThéorieAnalitiquede la Chaleur (1822)

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

  5. 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

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

  7. Complex numbers Complex number Its complex conjugate

  8. Complex numbers polar Complex number in polar coordinates

  9. Euler’s formula ? Sin (θ) ? Cos (θ)

  10. Im Re

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

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

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

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

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

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

  17. Fourier Series Periodic with period T Series of sines and cosines, see Euler’s formula with

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

  19. Fourier and Euler • Fourier • Euler

  20. 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

  21. Examples – Block 1 A -W/2 W/2

  22. Examples – Block 2

  23. Examples – Block 3 ?

  24. Examples – Impulse constant

  25. Examples – Shifted impulse Euler

  26. Examples – Shifted impulse 2 impulse constant Real part Imaginary part

  27. Also: using the following symmetry

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

  29. Examples - Impulse train 2

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

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

  32. Fourier + Convolution What is the Fourier domain equivalent of convolution?

  33. What is

  34. Intermezzo 1 What is ? Let , so

  35. Intermezzo 2 Property of Fourier Transform

  36. Fourier + Convolution cont’d

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

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

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

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

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

  42. Sampling - 2

  43. Sampling - 3

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

  45. FT of sampled functions

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

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

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

  49. Discrete Fourier Transform Continuous transform of sampled function

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