1. 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

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

3. Introduction Jean Baptiste Joseph Fourier (*1768-†1830)
French Mathematician
La Théorie Analitique de la Chaleur (1822)

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

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. What does this look like?
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. Series of sines and cosines, see Euler’s formula
Fourier Series
Periodic with period T
with
Series of sines and cosines, see Euler’s formula

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
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. Intermezzo: Symmetry in the FT

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

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

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

35. What is

36. Intermezzo 1
What is ?
Let , so

37. Intermezzo 2
Property of Fourier Transform

38. Fourier + Convolution cont’d

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

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

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

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

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

44. Sampling - 2

45. Sampling - 3

46. Sampling - 4

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

48. FT of sampled functions

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

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

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

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

53. 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.

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

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

56. Aliasing
If , aliasing can occur

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

58. Discrete Fourier Transform
Continuous transform of sampled function

59. is continuous and infinitely periodic with period 1/ΔT

60. 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

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

62. By now we probably need some …