1. System Arch 2007 (Fire Tom Wada)
OUTLINE
Wave
Sinusoidal signal review
Complex exponential signal
Complex amplitude (phasor)
What is Spectrum
2019/4/23
System Arch (Fire Tom Wada)

2. System Arch 2007 (Fire Tom Wada)
2 D wave
POND
BOAT person can send message by making water wave. LAND person can understand message by the movement of the float.
2019/4/23
System Arch (Fire Tom Wada)

3. System Arch 2007 (Fire Tom Wada)
Even with FOG
POND
LAND person just observe ONLY the float.
2019/4/23
System Arch (Fire Tom Wada)

4. System Arch 2007 (Fire Tom Wada)
Even with NOISE
POND
If main signal is enough stronger than the interference (noise),
LAND person can observe the message.
2019/4/23
System Arch (Fire Tom Wada)

5. Typical wave “Sinusoids”
t : time
A : amplitude
ω0 : radian frequency
Φ: phase-shift
2019/4/23
System Arch (Fire Tom Wada)

6. Review of cosine and sineθ r x y
2019/4/23
System Arch (Fire Tom Wada)

7. System Arch 2007 (Fire Tom Wada)
Period T0 of sinusoids
2019/4/23
System Arch (Fire Tom Wada)

8. QUIZ – what is the equation?-
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System Arch (Fire Tom Wada)

9. System Arch 2007 (Fire Tom Wada)
Review of complex
Cartesian Form
Complex z=(x, y)
x = Real part of z :
y = Imaginary part of z :
z = x + j y
Polar Form
Complex z=(r, θ)
2019/4/23
System Arch (Fire Tom Wada)

10. Cartesian Form and Polar Form
z=(x, y) y y r θ x x
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System Arch (Fire Tom Wada)

11. System Arch 2007 (Fire Tom Wada)
Transform
Polar to Cartesian
Cartesian to Polar
2019/4/23
System Arch (Fire Tom Wada)

12. System Arch 2007 (Fire Tom Wada)
Euler’s law
r=1 θ
2019/4/23
System Arch (Fire Tom Wada)

13. Complex exponential signal (Rotation Function)
Amplitude = A
Θ＝ω0t + Φ
Φ: phase-shift
ω0 : rotation speed
+ : counterclockwise
- : clockwise A -A
ω0 radian per second
2019/4/23
System Arch (Fire Tom Wada)

14. System Arch 2007 (Fire Tom Wada)
The complex exponential signal is another representation of cosine signal
Re means a Projection to X-axis
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System Arch (Fire Tom Wada)

15. System Arch 2007 (Fire Tom Wada)
Inverse Euler’s law
X* is conjugate of X.
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System Arch (Fire Tom Wada)

16. System Arch 2007 (Fire Tom Wada)
Inverse Euler’s law (2)
Real signal is the combination of + and - frequency complex exponential signal.
2019/4/23
System Arch (Fire Tom Wada)

17. Complex amplitude (Phasor)
X is called as Complex Amplitude or Phasor.
2019/4/23
System Arch (Fire Tom Wada)

18. Complex amplitude (Phasor) -2-1 -1
ω0 radian per second A -A
ω0 radian per second
X shows the t=0 position.
2019/4/23
System Arch (Fire Tom Wada)

19. System Arch 2007 (Fire Tom Wada)
Why Phasor is useful?
If you want to add two waves….
Make the complex exponential signal
2019/4/23
System Arch (Fire Tom Wada)

20. System Arch 2007 (Fire Tom Wada)
Why Phasor is useful? (2)
Make the Phasor…
Add them…
2019/4/23
System Arch (Fire Tom Wada)

21. System Arch 2007 (Fire Tom Wada)
Why Phasor is useful? (3)
Result
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System Arch (Fire Tom Wada)

22. 2 different frequency composition
2 apart frequencies
2 close frequencies
Beat
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System Arch (Fire Tom Wada)

23. System Arch 2007 (Fire Tom Wada)
Sum of N cosine waves
This is sum of 2N+1 different frequency signals.
2019/4/23
System Arch (Fire Tom Wada)

24. System Arch 2007 (Fire Tom Wada)
What is Spectrum
Spectrum is the set of phasors for each frequencies.
2019/4/23
System Arch (Fire Tom Wada)

25. System Arch 2007 (Fire Tom Wada)
Spectrum plot
N=3
frequency f (Hz)
-f3
-f2
-f1 f1 f2 f3
The line length is proportional to |phasor|.
2019/4/23
System Arch (Fire Tom Wada)

26. System Arch 2007 (Fire Tom Wada)
Example
frequency f (Hz)
-250
-100
100
250
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System Arch (Fire Tom Wada)

27. System Arch 2007 (Fire Tom Wada)
HW1-1
2.30
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System Arch (Fire Tom Wada)

28. System Arch 2007 (Fire Tom Wada)
HW1-2
2.68
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System Arch (Fire Tom Wada)

29. System Arch 2007 (Fire Tom Wada)
HW1-3
3.9
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System Arch (Fire Tom Wada)

30. System Arch 2007 (Fire Tom Wada)
References
DSP First – A Multimedia Approach
2019/4/23
System Arch (Fire Tom Wada)