1. HO#2 System Arch 200７ (Fire Tom Wada)
OUTLINE
Periodic Signal
Fourier series introduction
Sinusoids Orthogonality
Integration vs inner product
2019/5/6
HO#2 System Arch 200７ (Fire Tom Wada)

2. Consider any wave is sum of simple sin and cosine
Periodic Tc
2019/5/6
HO#2 System Arch 200７ (Fire Tom Wada)

3. HO#2 System Arch 200７ (Fire Tom Wada)
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
2019/5/6
HO#2 System Arch 200７ (Fire Tom Wada)

4. Spectrum of periodic signal
frequency f (Hz)
-5・fc
-4・fc
-3・fc
-2・fc
-fc fc
2・fc
3・fc
4・fc
5・fc
There are only n * fc (n=integer) frequencies!
2019/5/6
HO#2 System Arch 200７ (Fire Tom Wada)

5. Another example (even rectangular pulse)
2019/5/6
HO#2 System Arch 200７ (Fire Tom Wada)

6. Increase the number of sum (1)
2019/5/6
HO#2 System Arch 200７ (Fire Tom Wada)

7. Increase the number of sum (2)
2019/5/6
HO#2 System Arch 200７ (Fire Tom Wada)

8. HO#2 System Arch 200７ (Fire Tom Wada)
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.
2019/5/6
HO#2 System Arch 200７ (Fire Tom Wada)

9. HO#2 System Arch 200７ (Fire Tom Wada)
Fourier Series
If f(t) ‘s period is Tc…
If we use complex exponential…,
2019/5/6
HO#2 System Arch 200７ (Fire Tom Wada)

10. HO#2 System Arch 200７ (Fire Tom Wada)
Anyway, when you see the periodic signal, Please think it is just sum of sinusoids!!!
2019/5/6
HO#2 System Arch 200７ (Fire Tom Wada)

11. How we can divide f(t) into sinusoids?
Filter
Pass
nω (Hz)
Filter is used
an and bn
2019/5/6
HO#2 System Arch 200７ (Fire Tom Wada)

12. If we integrate in [ 0 to Tc]
2019/5/6
HO#2 System Arch 200７ (Fire Tom Wada)

13. If we integrate in [ 0 to Tc] (2)
a1 can be computed
2019/5/6
HO#2 System Arch 200７ (Fire Tom Wada)

14. If we integrate in [ 0 to Tc] (3)
b1 can be computed
2019/5/6
HO#2 System Arch 200７ (Fire Tom Wada)

15. By changing multiplier, each coefficient computed
One coefficient
2019/5/6
HO#2 System Arch 200７ (Fire Tom Wada)

16. Sinusoidal Orthogonality
m,n: integer, Tc=1/f0
Orthogonal
Orthogonal
Orthogonal
2019/5/6
HO#2 System Arch 200７ (Fire Tom Wada)

17. Another Orthogonality (1)
Vector inner product
Orthogonal
Θ＝90 degree
2019/5/6
HO#2 System Arch 200７ (Fire Tom Wada)

18. Another Orthogonality (2)
n dimensional vector IF
THEN A and B are Orthogonal.
2019/5/6
HO#2 System Arch 200７ (Fire Tom Wada)

19. is same as the N dim inner product
Freq=nω(Hz) sinusoids are Orthogonal each other (n=integer)
2019/5/6
HO#2 System Arch 200７ (Fire Tom Wada)

20. Fourier Series Summary
2019/5/6
HO#2 System Arch 200７ (Fire Tom Wada)

21. Complex form Fourier Series
Orthogonal
2019/5/6
HO#2 System Arch 200７ (Fire Tom Wada)

22. HO#2 System Arch 200７ (Fire Tom Wada)
HW2
[2-1]Compute the complex form Fourier Series coefficient cn for f(x).
[2-2]Draw the Spectrum of f(t) when T0=0.04sec.
2.30
2019/5/6
HO#2 System Arch 200７ (Fire Tom Wada)