Lecture 2: Fourier Series

1. SignalsandSpectral Methods in Geoinformatics Lecture 2: Fourier Series

2. Development of a function defined in an interval intoFourier Series Jean Baptiste Joseph Fourier

3. REPRESENTING A FUNCTION BY NUMBERS f(t) t 0 Τ coefficientsα1, α2, ... of the function f = a1φ1+ a2 φ2 + ... known base functions φ1, φ2, ... functionf

4. The base functions ofFourier series +1 0 –1 0 Τ +1 +1 +1 +1 0 0 0 0 –1 –1 –1 –1 0 Τ 0 Τ 0 Τ 0 Τ +1 +1 +1 +1 0 0 0 0 –1 –1 –1 –1 0 Τ 0 Τ 0 0 Τ Τ

5. Development of a real functionf(t)defined in the interval[0,T]intoFourier series 3 alternative forms:

6. Development of a real functionf(t)defined in the interval[0,T]intoFourier series 3 alternative forms: Everybase functionhas: period:

7. Development of a real functionf(t)defined in the interval[0,T]intoFourier series 3 alternative forms: Everybase functionhas: period: frequency:

8. Development of a real functionf(t)defined in the interval[0,T]intoFourier series 3 alternative forms: Everybase functionhas: period: frequency: angular frequency:

9. Development of a real functionf(t)defined in the interval[0,T]intoFourier series 3 alternative forms: Everybase functionhas: period: frequency: angular frequency: fundamentalperiod fundamental frequency fundamental angular frequency

10. Development of a real functionf(t)defined in the interval[0,T]intoFourier series 3 alternative forms: Everybase functionhas: period: frequency: angular frequency: fundamentalperiod fundamental frequency fundamental angular frequency term periods term frequencies term angular frequencies

11. Development of a real functionf(t)defined in the interval[0,T]intoFourier series simplest form:

12. Development of a real functionf(t)defined in the interval[0,T]intoFourier series simplest form: Fourier basis (base functions):

13. Development of a real functionf(t)defined in the interval[0,T]intoFourier series simplest form: Fourier basis (base functions):

14. +1 f(x) 0 –1 An example for the development of a function inFourier series Separate analysis of each term fork = 0, 1, 2, 3, 4, …

15. +1 +1 f(x) 0 0 –1 –1 k = 0 base function

16. +1 +1 f(x) 0 0 –1 –1 k = 0 contribution of term

17. +1 +1 +1 f(x) 0 0 0 –1 –1 –1 k = 1 base functions

18. +1 +1 +1 f(x) 0 0 0 –1 –1 –1 k = 1 contributions of term

19. +1 +1 +1 f(x) 0 0 0 –1 –1 –1 k = 2 base functions

20. +1 +1 +1 f(x) 0 0 0 –1 –1 –1 k = 2 contributions of term

21. +1 +1 +1 f(x) 0 0 0 –1 –1 –1 k = 3 base functions

22. +1 +1 +1 f(x) 0 0 0 –1 –1 –1 k = 3 contributions of term

23. +1 +1 +1 f(x) 0 0 0 –1 –1 –1 k = 4 base functions

24. +1 +1 +1 f(x) 0 0 0 –1 –1 –1 k = 4 contributions of term

25. +1 f(t) 0 –1

26. Exploiting the idea of function othogonality vector: orthogonal vector basis inner product:

27. Exploiting the idea of function othogonality vector: orthogonal vector basis inner product: Computation of vector components:

28. Exploiting the idea of function othogonality vector: orthogonal vector basis inner product: Computation of vector components: 0 0 0 0 0 0

29. Exploiting the idea of function othogonality vector: orthogonal vector basis inner product: Computation of vector components: 0 0 0 0 0 0

30. Exploiting the idea of function othogonality vector: orthogonal vector basis inner product: Computation of vector components: 0 0 0 0 0 0

31. Exploiting the idea of function othogonality vector: orthogonal vector basis inner product: Computation of vector components: 0 0 0 0 0 0

32. Orthogonality of Fourier base functions Inner product of two functions: Fourier basis: Norm (length) of a function: Orthogonality relations (km):

33. Computation of Fourier series coefficients Ortjhogonality relations (km):

34. Computation of Fourier series coefficients Ortjhogonality relations (km):

35. Computation of Fourier series coefficients Ortjhogonality relations (km): 0 0 0 0 forkm 0 0 0 0 forkm

36. Computation of Fourier series coefficients Ortjhogonality relations (km): 0 0 0 0 forkm 0 0 0 0 forkm

37. Computation of Fourier series coefficients Ortjhogonality relations (km): 0 0 0 0 forkm 0 0 0 0 forkm

38. Computation of Fourier series coefficients Ortjhogonality relations (km): 0 0 0 0 forkm 0 0 0 0 forkm

39. Computation of Fourier series coefficients Computation of Fourier series coefficients of a known function:

40. Computation of Fourier series coefficients

41. Computation of Fourier series coefficients change of notation

42. Computation of Fourier series coefficients change of notation

43. Computation of Fourier series coefficients

44. Computation of Fourier series coefficients

45. Alternative forms of Fourier series (polarforms) Polar coordinatesρk,θkorρk,φk, from the Cartesianak,bk ! ρk= «length» θk = «azimuth» φk+ θk= 90 φk = «direction angle»

46. Alternative forms of Fourier series (polarforms) Polar coordinatesρk,θkorρk,φk, from the Cartesianak,bk ! ρk= «length» θk = «azimuth» φk+ θk= 90 φk = «direction angle»

47. Alternative forms of Fourier series (polarforms) Polar coordinatesρk,θkorρk,φk, from the Cartesianak,bk ! ρk= «length» θk = «azimuth» φk+ θk= 90 φk = «direction angle»

48. Alternative forms of Fourier series (polarforms)

49. Alternative forms of Fourier series (polarforms)

50. Alternative forms of Fourier series (polarforms) θk = phase (sin)