Lecture 3: Fourier Transform

1. SignalsandSpectral Methods in Geoinformatics Lecture 3: Fourier Transform

2. Fourier transform and inverseFourier transform direct inverse

3. Fourier transform and inverseFourier transform direct from thenumber domain to thefrequency domain inverse

4. Fourier transform and inverseFourier transform direct from thefrequency domain to thenumber domain inverse

5. Fourier series in the interval [0, Τ] Fourier transform in the interval (-,+) inverse direct

6. FromFourierseries toFourier transform

7. FromFourierseries toFourier transform Change of notation

8. FromFourierseries toFourier transform Change of notation

9. FromFourierseries toFourier transform Change of notation

10. FromFourierseries toFourier transform

11. FromFourierseries toFourier transform

12. FromFourierseries toFourier transform

13. FromFourierseries toFourier transform

14. FromFourierseries toFourier transform direct inverse

15. Fourier series in a continuously increasing intervalΤ∞

16. Fourier series in a continuously increasing intervalΤ∞

17. Fourier series in a continuously increasing intervalΤ∞

18. Fourier series in a continuously increasing intervalΤ∞

19. Fourier series in a continuously increasing intervalΤ∞

20. Fourier series in a continuously increasing intervalΤ∞ 3

21. Fourier series in a continuously increasing intervalΤ∞

22. Fourier series in a continuously increasing intervalΤ∞ The Fourier series expansion of a function in a continuously larger interval Τ, provides coefficients for continuously denser frequencies ωk. As the length of the interval Τ tends to infinity,the frequencies ωktend to covermore and more from the set of the real values frequencies () For an infinite interval Τ, i.e. for (t)the total real set of frequencies () is required and fromthe Fourier series expansion we pass to the inverseFourier transform continuousfrequencies - all possible values () discretefrequenciesωk wit step Δω = 2π/Τ

23. Fouriertransform of a complex function

24. Fouriertransform of a complex function

25. Fouriertransform of a complex function

26. Fouriertransform of a complex function

27. Fouriertransform of a complex function

28. Fouriertransform of a complex function complex form real form

29. Fouriertransform of a complex function direct inverse Notation Usual (mathematicallyincorect) notation

30. Fourier transform of areal function Complex function:

31. Fourier transform of areal function Complex function: Real function:

32. Fourier transform of areal function Complex function: Real function:

33. Fourier transform of areal function Complex function: Real function: sine transform cosinetransform

34. Fourier transform of areal function Complex function: Real function: sine transform cosinetransform

35. Fourier transform of areal function even function odd function

36. Fouriertransform in polar form |F(ω)|

37. Fouriertransform in polar form phase spectrum amplitude spectrum polar form:

38. Fouriertransform in polar form even odd amplitude spectrum even function phase spectrum odd function

39. Properties of theFourier transform Linearity

40. Properties of theFourier transform Linearity Symmetry

41. Properties of theFourier transform Linearity Symmetry Time translation

42. Properties of theFourier transform Linearity Symmetry Time translation

43. Properties of theFourier transform Linearity Symmetry Time translation

44. Properties of theFourier transform Phase translation

45. Properties of theFourier transform Phase translation Modulation theorem

46. Properties of theFourier transform Modulation theorem Proof:

47. Properties of theFourier transform Modulation theorem Proof:

48. Properties of theFourier transform Modulation theorem Proof:

49. Properties of theFourier transform Modulation theorem Proof:

50. Properties of theFourier transform Modulation theorem Proof: