1. OUTLINE Wave Sinusoidal signal review Complex exponential signal
Complex amplitude (phasor)
What is Spectrum
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2. 2 D wave
POND
BOAT person can send message by making water wave. LAND person can understand message by the movement of the float.
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3. Even with FOG POND LAND person just observe ONLY the float. 2009/10/19
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4. Even with NOISE
POND
If main signal is enough stronger than the interference (noise),
LAND person can observe the message.
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5. Typical wave “Sinusoids”
t : time
A : amplitude
ω0 : radian frequency
Φ: phase-shift
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6. Review of cosine and sineθ r x y
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7. Period T0 of sinusoids
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8. QUIZ – what is the equation?-
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9. 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)
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10. Cartesian Form and Polar Form
z=(x, y) y y r θ x x
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11. Transform
Polar to Cartesian
Cartesian to Polar
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12. Euler’s law
r=1 θ
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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
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14. The complex exponential signal is another representation of cosine signal
Re means a Projection to X-axis
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15. Inverse Euler’s law
X* is conjugate of X.
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16. Inverse Euler’s law (2)
Real signal is the combination of + and - frequency complex exponential signal.
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17. Complex amplitude (Phasor)
X is called as Complex Amplitude or Phasor.
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18. Complex amplitude (Phasor) -2-1 -1
ω0 radian per second A -A
ω0 radian per second
X shows the t=0 position.
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19. Why Phasor is useful? If you want to add two waves….
Make the complex exponential signal
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20. Why Phasor is useful? (2) Make the Phasor… Add them… 2009/10/19
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21. Why Phasor is useful? (3)
Result
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22. 2 different frequency composition
2 apart frequencies
2 close frequencies
Beat
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23. Sum of N cosine waves This is sum of 2N+1 different frequency signals.
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24. What is Spectrum Spectrum is the set of phasors for each frequencies.
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25. Spectrum plot N=3 The line length is proportional to |phasor|.
frequency f (Hz)
-f3
-f2
-f1 f1 f2 f3
The line length is proportional to |phasor|.
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26. Example
frequency f (Hz)
-250
-100
100
250
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27. HW1-1
2.30
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28. HW1-2
2.68
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29. HW1-3
3.9
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30. HW submission Submit your Home Work to Ikenoya-san
Subject: SysArch09-HW1
Show your Student ID and Your Name
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31. References
DSP First – A Multimedia Approach
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