1. 2009/10/26 System Arch 1 OUTLINE Periodic Signal Fourier series introduction Sinusoids Orthogonality Integration vs inner product

2. 2009/10/26 System Arch 2 Consider any wave is sum of simple sin and cosine Periodic Tc

3. 2009/10/26 System Arch 3 Periodic signal is composed of DC + same frequency sinusoid + multiple frequency sinusoids Frequency = 0 Hz Basic frequency fc=1/Tc 2 x fc 3 x fc 4 x fc

4. 2009/10/26 System Arch 4 Spectrum of periodic signal frequency f (Hz) 0fc 2 ・ fc 3 ・ fc 4 ・ fc 5 ・ fc -fc -2 ・ fc -3 ・ fc -4 ・ fc -5 ・ fc There are only n * fc (n=integer) frequencies!

5. 2009/10/26 System Arch 5 Another example (even rectangular pulse)

6. 2009/10/26 System Arch 6 Increase the number of sum (1) N=1 N=2 N=3N=10

7. 2009/10/26 System Arch 7 Increase the number of sum (2) N=20 N=50 N=100N=200

8. 2009/10/26 System Arch 8 Fourier Jean Baptiste Joseph, Baron de Frourier France, 1778/Mar/21 – 1830/May/16 Fourier Series paper is written in 1807 Even discontinue function (such as rectangular pulse) can be composed of many sinusoids. Nobody believed the paper at that time.

9. 2009/10/26 System Arch 9 Fourier Series If f(t) ‘s period is Tc… If we use complex exponential…,

10. 2009/10/26 System Arch 10 Anyway, when you see the periodic signal, Please think it is just sum of sinusoids!!!

11. 2009/10/26 System Arch 11 How we can divide f(t) into sinusoids? Filter Pass nω (Hz) Filter is used a n and b n

12. 2009/10/26 System Arch 12 If we integrate in [ 0 to Tc] Tc

13. 2009/10/26 System Arch 13 If we integrate in [ 0 to Tc] (2) Tc a 1 can be computed

14. 2009/10/26 System Arch 14 If we integrate in [ 0 to Tc] (3) Tc b 1 can be computed

15. 2009/10/26 System Arch 15 By changing multiplier, each coefficient computed Tc One coefficient

16. 2009/10/26 System Arch 16 Sinusoidal Orthogonality m,n: integer, T c =1/f 0 Orthogonal

17. 2009/10/26 System Arch 17 Another Orthogonality (1) Vector inner product Orthogonal Θ ＝ 90 degree

18. 2009/10/26 System Arch 18 Another Orthogonality (2) n dimensional vector IF THEN A and B are Orthogonal.

19. 2009/10/26 System Arch 19 is same as the N dim inner product Freq=nω(Hz) sinusoids are Orthogonal each other (n=integer)

20. 2009/10/26 System Arch 20 Fourier Series Summary

21. 2009/10/26 System Arch 21 Complex form Fourier Series Orthogonal

22. 2009/10/26 System Arch 22 HW2 [2-1]Compute the complex form Fourier Series coefficient cn for f(x). [2-2]Draw the Spectrum of f(t) when T 0 =0.04sec. 2.30