1. Da-Chuan Cheng, PhD12015/4/13Da-Chuan Cheng, PhD1 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.

2. Da-Chuan Cheng, PhD2 2 Computation of Fourier series : Real form Important properties of Fourier series: (1) (2) (3)

3. Da-Chuan Cheng, PhD32015/4/13Da-Chuan Cheng, PhD3 Computation of Fourier series : Real form An equivalent way of starting this theorem is that the collection is an orthonormal set of functions in

4. Da-Chuan Cheng, PhD4 Matlab program Da-Chuan Cheng, PhD4 % 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

5. Da-Chuan Cheng, PhD5 5

6. 6 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

7. Da-Chuan Cheng, PhD7 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

8. Da-Chuan Cheng, PhD8 8 % Confirm cos(t)sin(t)=0 z = cos(t).*sin(t); sum(z) ans = 4.9960e-016 % confirm cos(t)*sin(2t)=0 z = cos(t).*sin(2*t); sum(z) ans = -1.1102e-015

9. Da-Chuan Cheng, PhD9 Computation of Fourier series : Real form Da-Chuan Cheng, PhD9 Proof of the properties on page 2: The derivations of the first two equalities use the following identities:

10. Da-Chuan Cheng, PhD102015/4/13Da-Chuan Cheng, PhD10 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))

11. Da-Chuan Cheng, PhD112015/4/13Da-Chuan Cheng, PhD11 Computation of Fourier series : Real form

12. Da-Chuan Cheng, PhD122015/4/13Da-Chuan Cheng, PhD12 Computation of Fourier series : Real form 實際上就是訊號 f(x) 的平均值

13. Da-Chuan Cheng, PhD132015/4/13Da-Chuan Cheng, PhD13 Computation of Fourier series : Real form Theorem: If then, The Fourier coefficients for a given function are unique.

14. Da-Chuan Cheng, PhD14 Matlab implementation Da-Chuan Cheng, PhD14 % 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

15. Da-Chuan Cheng, PhD15 Change the representation form see p.13 or

16. Da-Chuan Cheng, PhD16 Computation of Fourier series : Real form Theorem: If then,

17. Da-Chuan Cheng, PhD172015/4/13Da-Chuan Cheng, PhD17 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

18. Da-Chuan Cheng, PhD182015/4/13Da-Chuan Cheng, PhD18 Computation of Fourier series : Real form Theorem:

19. Da-Chuan Cheng, PhD192015/4/13Da-Chuan Cheng, PhD19 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.

20. Da-Chuan Cheng, PhD202015/4/13Da-Chuan Cheng, PhD20 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

21. Da-Chuan Cheng, PhD212015/4/13Da-Chuan Cheng, PhD21 Computation of Fourier series (2) : Complex form Lemma: Theorem: Proof:

22. Da-Chuan Cheng, PhD222015/4/13Da-Chuan Cheng, PhD22 Computation of Fourier series (3) : Complex form Theorem: Combine with the previous theorem in Chap2 p.7, we get the following theorem: Proof: To findwe simply substitute f(t) into: Use the properties in p.2.

23. Da-Chuan Cheng, PhD232015/4/13Da-Chuan Cheng, PhD23 Property (3) 0 0 (1) n≠m =0

24. Da-Chuan Cheng, PhD242015/4/13Da-Chuan Cheng, PhD24 (2) n=m=0 (3) n=m ≧ 1

25. Da-Chuan Cheng, PhD252015/4/13Da-Chuan Cheng, PhD25 Computation of Fourier series (4) : Complex form So the complex Fourier series of f is Example: The n-th complex Fourier coefficients is:

26. Da-Chuan Cheng, PhD262015/4/13Da-Chuan Cheng, PhD26 Computation of Fourier series (5) : Complex form 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); Matlab implementation

27. Da-Chuan Cheng, PhD272015/4/13Da-Chuan Cheng, PhD27 Computation of Fourier series (6) : Complex form Result: k=-10:10 real partimaginary part

28. Da-Chuan Cheng, PhD282015/4/13Da-Chuan Cheng, PhD28 Computation of Fourier series (7) : Complex form Result: k=-50:50 real partimaginary part

29. Da-Chuan Cheng, PhD292015/4/13Da-Chuan Cheng, PhD29 Computation of Fourier series (8) : Complex form Result: k=-100:100 real partimaginary part

30. Da-Chuan Cheng, PhD302015/4/13Da-Chuan Cheng, PhD30 Computation of Fourier series (9) : Complex form Result: k=-10000:10000 real partimaginary part

31. Da-Chuan Cheng, PhD312015/4/13Da-Chuan Cheng, PhD31 Computation of Fourier series (10) : Complex form Theorem: The set of functions Example:

32. Da-Chuan Cheng, PhD322015/4/13Da-Chuan Cheng, PhD32 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. If f is real valued, then n≠0

33. Da-Chuan Cheng, PhD332015/4/13Da-Chuan Cheng, PhD33 Computation of Fourier series (12) : Complex form

34. Da-Chuan Cheng, PhD342015/4/13Da-Chuan Cheng, PhD34 Computation of Fourier series (13) : Complex form similar to p. 13 (see p. 13)

35. Da-Chuan Cheng, PhD352015/4/13Da-Chuan Cheng, PhD35 Computation of Fourier series (14) : Complex form Exactly the same to page 13.

36. Da-Chuan Cheng, PhD36 Homework 請自行找出一個長 32 的信號﹐寫 Matlab 程式 將其 Fourier series 的係數 (a 0, a n, b n ) 找出。 Issue date: 5/5 Due date: 19/5