1. Fourier Transformation
Transformasjon
f(x)
F(u)

2. Continuous Fourier Transform Def
The Fourier transform
of a one-dimentional function f(x)
The Inverse Fourier Transform

3. Continuous Fourier Transform Def - Notation
The Fourier transform
of a one-dimentional function f(x)
The inverse Fourier Transform of F(u)

4. Continuous Fourier Transform Alternative Def

5. Continuous Fourier Transform Example - cos(2ft)

6. Continuous Fourier Transform Example - cos(t)

7. Continuous Fourier Transform Example - sin(t)

8. Continuous Fourier Transform Example - Delta-function

9. Continuous Fourier Transform Example - Gauss function

10. Signals and Fourier Transform Frequency InformationFT FT
Øverst vises funksjonen y1 = sin(w1*t), dvs en funksjon med en gitt frekvens w1.
Den Fourier-transformerte funksjonen viser en enkelt topp svarende til denne ene frekvensen.
I midten vises funksjonen y2 = sin(w2*t) med en gitt frekvens w2 hvor w2 > w1.
Den Fourier-transformerte funksjonen viser en enkelt topp svarende til denne ene frekvensen, men vi ser at denne toppen er plassert lenger til høyre
enn den toppen i den forrige Fourier-transformerte, svarende til at vi nå har en høyere frekvens.
Nederst vises en funksjon y3 = sin(w1*t) + sin(w2*t), dvs en funksjon som inneholder to gitte frekvenser.
Dette gjenspeiles i den Fourier-transformerte som to topper i diagrammet, de to toppene fra de to foregående Fourier-transformerte. FT

11. Stationary / Non-stationary signalsFT
Non stationary
Øverst vises en funksjon y3 = sin(w1*t) + sin(w2*t), dvs en funksjon som inneholder to gitte frekvenser (samme som siste figur på foregående slide),
samt den Fourier-transformerte som inneholder to topper svarende til de to frekvensene.
Legg merke til at de to frekvensene opptrer samtidig hele tiden i tid-rommet.
Nederst vises en grein-funksjon y4 hvor de to frekvensene ikke lengre opptrer samtidig i tid.
Først opptrer frekvensen w1, deretter overtar frekvensen w2.
Legg merke til at De to Fourier-transformerte funksjonene er like, dvs en Fourier-transformasjon kan plukke ut de enkelte frekvensene, men klarer ikke å plassere
disse i tid. FT
The stationary and the non-stationary signal both have the same FT.
FT is not suitable to take care of non-stationary signals to give information about time.

12. Transient Signal Frequency Information
Constant function in [-3,3].
Dominating frequency  = 0
and some freequency because of edges.
Transient signal
resulting in extra frequencies > 0.
Narrower transient signal
resulting in extra higher frequencies
pushed away from origin.

13. Transient Signal No Information about Position
Moving the transient part of the signal to a new position does not result
in any change in the transformed signal.
Conclusion: The Fourier transformation
contains information of a transient part
of a signal, but only the frequency
not the position.

14. Inverse Fourier Transform [1/3]
Theorem:
Proof:

15. Inverse Fourier Transform [2/3]
Theorem:
Proof:

16. Inverse Fourier Transform [3/3]

17. Properties

18. Fourier Transforms of Harmonic and Constant Function

19. Fourier Transforms of Some Common Functions

20. Even and Odd Functions [1/3]
Def
Every function can be split
in an even and an odd part
Every function can be split
in an even and an odd part
and each of this can in turn be split in a real and an imaginary part

21. Even and Odd Functions [2/3]
1. Even component in f produces an even component in F
2. Odd component in f produces an odd component in F
3. Odd component in f produces an coefficient -j

22. Even and Odd Functions [3/3]

23. The Shift Theorem

24. The Similarity Theorem

25. The Convolution Theorem

26. Convolution Edge detection

27. The Adjoint of the Fourier Transform
Theorem: Suppose f and g er are square integrable. Then:
Proof:

28. Plancherel Formel - The Parselval’s Theorem
Theorem: Suppose f and g are square integrable. Then:
Proof:

29. The Rayleigh’s Theorem Conservation of Energy
The energy of a signal in the time domain
is the same as the energy in the frequency domain

30. The Fourier Series Expansion u a discrete variable - Forward transform
Suppose f(t) is a transient function that is zero outside the interval [-T/2,T/2] or is considered to be one cycle of a periodic function.
We can obtain a sequence of coefficients by making a discrete variable
and integrating only over the interval.

31. The Fourier Series Expansion u a discrete variable - Inverse transform
The inverse transform becomes:

32. The Fourier Series Expansion cn coefficients

33. The Fourier Series Expansion zn, an, bn coefficients

34. The Fourier Series Expansion an,bn coefficients

35. Fourier Series Pulse train
approximated by Fourier Serie
N = 1
N = 2
N = 5
N = 10

36. Fourier Series Pulse train – Java program

37. Pulse Train approximated by Fourier Serie
f(x) square wave (T=2)
N=1
N=2
N=10

38. Fourier Series Zig tag Zig tag approximated by Fourier Serie N = 1

39. Fourier Series Negative sinus function
approximated by Fourier Serie
N = 1
N = 2
N = 5
N = 10

40. Fourier Series Truncated sinus function
approximated by Fourier Serie
N = 1
N = 2
N = 5
N = 10

41. Fourier Series Line Line approximated by Fourier Serie N = 1 N = 2

42. Fourier Series Java program for approximating Fourier coefficients
Approximate functions by adjusting Fourier coefficients (Java program)

43. The Discrete Fourier Transform - DFT Discrete Fourier Transform - Discretize both time and frequency
Continuous
Fourier transform
Discrete frequency
Fourier Serie
Discrete frequency and time
Discrete Fourier Transform

44. The Discrete Fourier Transform - DFT Discrete Fourier Transform - Discretize both time and frequency
{ fi } sequence of length N,
taking samples of a continuous function at equal intervals

45. Continuous Fourier Transform in two Dimensions Def
The Fourier transform
of a two-dimentional function f(x,y)
The Inverse Fourier Transform

46. The Two-Dimensional DFT and Its Inverse

47. Fourier Transform in Two Dimensions Example 1

48. Fourier Transform in Two Dimensions Example 2

49. End