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# Chapter 3: Fourier Series

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1. Chapter 3: Fourier Series • Introduction • In 1808, Fourier wrote the first version of his celebrated memoir on the theory of heat “Theorie Analytique de la Chaleur”. He made a detailed study of trigonometric series, which he used to solve a variety of heat conduction problems. • Nearly two centuries after Fourier’s work, the series that bears his name is still important, practically and theoretically, and still a topic of current research. Da-Chuan Cheng, PhD Da-Chuan Cheng, PhD 1

2. Computation of Fourier series : Real form Important properties of Fourier series: (1) (2) (3) Da-Chuan Cheng, PhD Da-Chuan Cheng, PhD 2

3. Computation of Fourier series : Real form An equivalent way of starting this theorem is that the collection is an orthonormal set of functions in Da-Chuan Cheng, PhD Da-Chuan Cheng, PhD 3

4. Matlab program % Confirm page 3. L=37; % L can be any positive integer. t=[-pi:(2*pi)/L:pi]; % range is from -pi to (pi- dt) t=t(1:end-1); y=cos(t)/sqrt(L/2); % equation in page 3. y=cos(t). disp(['The length of vector y=cos(t) is : ']); sum(y.^2) % confirm the length=1. figure(1); plot(t,y); hold on; 長度不是π﹐而是L/2 The length of vector y=cos(t) is : ans = 1.0000 Da-Chuan Cheng, PhD Da-Chuan Cheng, PhD 4

5. Da-Chuan Cheng, PhD Da-Chuan Cheng, PhD 5

6. z1=sin(t)/sqrt(L/2); % equation in page 3. y=sin(t). disp(['The length of vector y=sin(t) is : ']); sum(z1.^2) % confirm the length=1. figure(1); plot(t,z1); hold on; The length of vector y=sin(t) is : ans = 1.0000 Da-Chuan Cheng, PhD Da-Chuan Cheng, PhD 6

7. y1=cos(2*t)/sqrt(L/2); % y=cos(2t). disp(['The length of vector y1=cos(2t) is : ']); sum(y1.^2) % confirm the length=1. figure(1); plot(t,y1,'y'); The length of vector y1=cos(2t) is : ans = 1 Da-Chuan Cheng, PhD Da-Chuan Cheng, PhD 7

8. % Confirm cos(t)sin(t)=0 z = cos(t).*sin(t); sum(z) % confirm cos(t)*sin(2t)=0 z = cos(t).*sin(2*t); sum(z) ans = 4.9960e-016 ans = -1.1102e-015 Da-Chuan Cheng, PhD Da-Chuan Cheng, PhD 8

9. Computation of Fourier series : Real form Proof of the properties on page 2: The derivations of the first two equalities use the following identities: Da-Chuan Cheng, PhD Da-Chuan Cheng, PhD 9

10. Computation of Fourier series : Real form Equation (2) and (3) can be proved in a similar way. Fourier coefficients computation: Assume Note that k starts from 1, why? (Hint: see page 2, Eq.(1)) Da-Chuan Cheng, PhD Da-Chuan Cheng, PhD 10

11. Computation of Fourier series : Real form Da-Chuan Cheng, PhD Da-Chuan Cheng, PhD 11

12. Computation of Fourier series : Real form 實際上就是訊號f(x)的平均值 Da-Chuan Cheng, PhD Da-Chuan Cheng, PhD 12

13. Computation of Fourier series : Real form Theorem: If then, The Fourier coefficients for a given function are unique. Da-Chuan Cheng, PhD Da-Chuan Cheng, PhD 13

14. Matlab implementation % Page 13. L=64; % L can be any positive integer. t=[-pi:(2*pi)/L:pi]; % range is from -pi to (pi- dt) t=t(1:end-1); f=rand(1,L).*sin(rand(1,L)*2); figure; plot(t,f); a0=mean(f) a1=sum(f.*cos(t))/L/2 b1=sum(f.*sin(t))/L/2 Da-Chuan Cheng, PhD Da-Chuan Cheng, PhD 14

15. Change the representation form see p.13 or Da-Chuan Cheng, PhD

16. Computation of Fourier series : Real form Theorem: If then, Da-Chuan Cheng, PhD

17. Computation of Fourier series : Real form Even and odd functions The following properties follow from the definition. Even X Even = Even Even X Odd = Odd Odd X Odd = Even If F is an even function, then If F is an odd function, then Da-Chuan Cheng, PhD Da-Chuan Cheng, PhD 17

18. Computation of Fourier series : Real form Theorem: Da-Chuan Cheng, PhD Da-Chuan Cheng, PhD 18

19. Computation of Fourier series : Real form Gibbs phenomenon Approximation of square wave in 5 steps The height of the blip is approximately the same no matter how many terms are considered in the partial sum. Da-Chuan Cheng, PhD Da-Chuan Cheng, PhD 19

20. Computation of Fourier series (1) :Complex form Complex form of Fourier Series Often, it is more convenient to express Fourier series in complex form using the complex exponentials due to the simple computational properties of these functions. Definition: This definition is motivated by substituting x=it into the Taylor series for Da-Chuan Cheng, PhD Da-Chuan Cheng, PhD 20

21. Computation of Fourier series (2) :Complex form Lemma: Theorem: Proof: Da-Chuan Cheng, PhD Da-Chuan Cheng, PhD 21

22. Computation of Fourier series (3) :Complex form Combine with the previous theorem in Chap2 p.7, we get the following theorem: Theorem: Proof: To find we simply substitute f(t) into: Use the properties in p.2. Da-Chuan Cheng, PhD Da-Chuan Cheng, PhD 22

23. 0 0 Property (3) (1) n≠m =0 Da-Chuan Cheng, PhD Da-Chuan Cheng, PhD 23

24. (2) n=m=0 (3) n=m≧1 Da-Chuan Cheng, PhD Da-Chuan Cheng, PhD 24

25. Computation of Fourier series (4) :Complex form Example: The n-th complex Fourier coefficients is: So the complex Fourier series of f is Da-Chuan Cheng, PhD Da-Chuan Cheng, PhD 25

26. Computation of Fourier series (5) :Complex form Matlab implementation t=[-pi:2*pi/1024:pi]; t=t(1:end-1); fs=zeros(1,length(t)); N=100; K=[-N:N]; for k=K; fs=fs+2/((2*k+1)*pi)*sin((2*k+1)*t); end; figure(1); subplot(121); plot(t,fs); fc=zeros(1,length(t)); for k=K, fc=fc-2/((2*k+1)*pi)*cos((2*k+1)*t); end; figure(1); subplot(122); plot(t,fc); Da-Chuan Cheng, PhD Da-Chuan Cheng, PhD 26

27. Computation of Fourier series (6) :Complex form Result: k=-10:10 real part imaginary part Da-Chuan Cheng, PhD Da-Chuan Cheng, PhD 27

28. Computation of Fourier series (7) :Complex form Result: k=-50:50 real part imaginary part Da-Chuan Cheng, PhD Da-Chuan Cheng, PhD 28

29. Computation of Fourier series (8) :Complex form Result: k=-100:100 real part imaginary part Da-Chuan Cheng, PhD Da-Chuan Cheng, PhD 29

30. Computation of Fourier series (9) :Complex form Result: k=-10000:10000 real part imaginary part Da-Chuan Cheng, PhD Da-Chuan Cheng, PhD 30

31. Computation of Fourier series (10) :Complex form Theorem: The set of functions Example: Da-Chuan Cheng, PhD Da-Chuan Cheng, PhD 31

32. Computation of Fourier series (11) :Complex form Relation between the real and complex Fourier series If f is a real valued function, the real form of its Fourier series can be derived from its complex form and vice versa. For simplicity, we discuss this derivation on the interval -π≦t ≦ π, but this discussion also holds for other intervals as well. n≠0 If f is real valued, then Da-Chuan Cheng, PhD Da-Chuan Cheng, PhD 32

33. Computation of Fourier series (12) :Complex form Da-Chuan Cheng, PhD Da-Chuan Cheng, PhD 33

34. Computation of Fourier series (13) :Complex form similar to p. 13 (see p. 13) Da-Chuan Cheng, PhD Da-Chuan Cheng, PhD 34

35. Computation of Fourier series (14) :Complex form Exactly the same to page 13. Da-Chuan Cheng, PhD Da-Chuan Cheng, PhD 35

36. Homework • 請自行找出一個長32的信號﹐寫Matlab程式將其Fourier series的係數(a0, an, bn)找出。 • Issue date: 5/5 • Due date: 19/5 Da-Chuan Cheng, PhD