1. 中華大學 資訊工程系 Fall 2002 Chap 5 Fourier Series

2. Page 2 Fourier Analysis Fourier Series Fourier Series Fourier Integral Fourier Integral Discrete Fourier Transform Discrete Fourier Transform Fourier Transform Fourier Transform Fast Fourier Transform Fast Fourier Transform Discrete Continuous

3. Page 3 Outline Periodic Function Fourier Cosine and Sine Series Periodic Function with Period 2L Odd and Even Functions Half Range Fourier Cosine and Sine Series Complex Notation for Fourier Series

4. Page 4 Fourier, Joseph 1768-1830

5. Page 5 Fourier, Joseph In 1807, Fourier submitted a paper to the Academy of Sciences of Paris. In it he derived the heat equation and proposed his separation of variables method of solution. The paper, evaluated by Laplace, Lagrange, and Lagendre, was rejected for lack rigor. However, the results were promising enough for the academy to include the problem of describing heat conduction in a prize competition in 1812. Fourier’s 1811 revision of his earlier paper won the prize, but suffered the same criticism as before. In 1822, Fourier finally published his classic Theorie analytique de la chaleur, laying the fundations not only for the separation of variables method and Fourier series, but for the Fourier integral and transform as well.

6. Page 6 Periodic Function Definition: Periodic Function A function f(x) is said to be periodic with period T if for all x T f(x) x

7. Page 7 Periodic Function f(x+p)=f(x), f(x+np)=f(x) If f(x) and g(x) have period p, the the function H(x)=af(x)+bg(x), also has the period p If a period function of f(x) has a smallest period p (p >0), this is often called the fundamental period of f(x)

8. Page 8 Periodic Function Example Cosine Functions: cosx, cos2x, cos3x, … Sine Functions: sinx, sin2x, sin3x, … e ix, e i2x, e i3x, … e -ix, e -i2x, e -i3x, …

9. Page 9 Fourier Cosine and Sine Series Lemma: Trigonometric System is Orthogonal

10. Page 10 Fourier Cosine and Sine Series A function f(x) is periodic with period 2  and

11. Page 11 Fourier Cosine and Sine Series (Euler formulas) Then

12. Page 12 Fourier Cosine and Sine Series Proof:

13. Page 13 Fourier Cosine and Sine Series Example 5-1: Find the Fourier coefficients corresponding to the function Sol:

14. Page 14 Fourier Cosine and Sine Series Sol:

15. Page 15 Fourier Cosine and Sine Series Sol:

16. Page 16 Periodic Function with Period 2L A periodic function f(x) with period 2L 2L f(x) x

17. Page 17 Periodic Function with Period 2L Then

18. Page 18 Periodic Square Wave

19. Page 19 Odd and Even Functions A function f(x) is said to be even if A function f(x) is said to be odd if

20. Page 20 Odd and Even Functions f(x) x x Even Function Odd Function

21. Page 21 Odd and Even Functions Property The product of an even and an odd function is odd.

22. Page 22 Odd and Even Functions Fourier Cosine Series Fourier Sine Series

23. Page 23 Sun of Functions The Fourier coefficients of a sum f 1 +f 2 are the sum of the corresponding Fourier coefficients of f 1 and f 2. The Fourier coefficients of cf are c times the corresponding Fourier coefficients of f.

24. Page 24 Examples Rectangular Pulse The function f*(x) is the sum in Example 1 of Sec.10.2 and the constant k. Sawtooth wave Find the Fourier series of the function

25. Page 25 Half-Range Expansions A function f is given only on half the range, half the interval of periodicity of length 2L. even periodic extention f 1 of f odd period extention f 2 of f

26. Page 26 Complex Notation for Fourier Series

27. Page 27 Complex Notation for Fourier Series A periodic function f(x) with period 2L

28. Page 28 Complex Fourier Series Find complex Fourier series

29. Page 29 Exercise Section 10-4 #1, Section 10-2 #5, #11 Section 10-3 #5, #9 Section 10-4 #1, #15

30. Page 30 Fourier Cosine and Sine Integrals Example 1

31. Page 31 Fourier Cosine and Sine Integrals x sin(x)

32. Page 32 Fourier Cosine and Sine Integrals x

33. Page 33 Fourier Cosine and Sine Integrals

34. Page 34 Fourier Cosine and Sine Integrals

35. Page 35 Fourier Cosine and Sine Integrals

36. Page 36 Gibb’s Phenomenon Sine Integral

37. Page 37 Gibb’s Phenomenon

38. Page 38 Gibb’s Phenomenon

39. Page 39 Fourier Cosine and Sine Integrals Fourier Cosine Integral of f(t)

40. Page 40 Fourier Cosine and Sine Integrals Fourier Sine Integral of f(t)

41. Page 41 Fourier Integrals Fourier Integral of f(t)