OUTLINE

1
OUTLINE

• Wave
• Sinusoidal signal review
• Complex exponential signal
• Complex amplitude (phasor)
• What is Spectrum

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2 D wave
BOAT person can send message by making water
wave.LAND person can understand message by the
movement of the float.
3
Even with FOG
LAND person just observe ONLY the float.
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Even with NOISE
If main signal is enough stronger than the
interference (noise), LAND person can observe the
message.
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Typical wave Sinusoids

• t time
• A amplitude
• ?0 radian frequency
• F phase-shift

6
Review of cosine and sine
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Period T0 of sinusoids
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QUIZ what is the equation?-
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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, ?)

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Cartesian Form and Polar Form
z(x, y)
y
y
r
?
x
x
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Transform

• Polar to Cartesian

• Cartesian to Polar

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Eulers law
r1
?
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Complex exponential signal (Rotation Function)

• Amplitude A
• T?0t F
• F phase-shift
• ?0 rotation speed
• counterclockwise
• - clockwise

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The complex exponential signal is another
representation of cosine signal

• Re means a Projection to X-axis

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Inverse Eulers law
X is conjugate of X.
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Inverse Eulers law (2)

• Real signal is the combination of and -
  frequency complex exponential signal.

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Complex amplitude (Phasor)

• X is called as Complex Amplitude or Phasor.

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Complex amplitude (Phasor) -2-

• X shows the t0 position.

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Why Phasor is useful?

1. If you want to add two waves.

1. Make the complex exponential signal

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Why Phasor is useful? (2)

1. Make the Phasor

1. Add them

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Why Phasor is useful? (3)

1. Result

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2 different frequency composition

• 2 apart frequencies

• 2 close frequencies
• Beat

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Sum of N cosine waves

• This is sum of 2N1 different frequency signals.

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What is Spectrum

• Spectrum is the set of phasors for each
  frequencies.

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Spectrum plot
N3
frequency f (Hz)
0
f1
-f1
f2
-f2
f3
-f3
The line length is proportional to phasor.
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Example
frequency f (Hz)
0
100
-100
250
-250
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HW1-1
2.30
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HW1-2
2.68
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HW1-3
3.9
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HW submission

• Submit your Home Work to kano-san
• kano_at_lsi.ie.u-ryukyu.ac.jp
• Subject SysArch10-HW1
• Show your Student ID and Your Name

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References

1. DSP First A Multimedia Approach