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System Arch 2006 (Fire Tom Wada)
OUTLINE Wave Sinusoidal signal review Complex exponential signal Complex amplitude (phasor) What is Spectrum HO# /April/21 System Arch (Fire Tom Wada)
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System Arch 2006 (Fire Tom Wada)
20 40 60 80 100 120 -2 -1 1 2 Real(1*exp(2πjx/32) Real(( j)*exp(2πjx/32) + || Real(( j)*exp(2πjx/32) HO# /April/21 System Arch (Fire Tom Wada)
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System Arch 2006 (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. HO# /April/21 System Arch (Fire Tom Wada)
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System Arch 2006 (Fire Tom Wada)
Even with FOG POND LAND person just observe ONLY the float. HO# /April/21 System Arch (Fire Tom Wada)
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System Arch 2006 (Fire Tom Wada)
Even with NOISE POND If main signal is enough stronger than the interference (noise), LAND person can observe the message. HO# /April/21 System Arch (Fire Tom Wada)
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Typical wave “Sinusoids”
t : time A : amplitude ω0 : radian frequency Φ: phase-shift HO# /April/21 System Arch (Fire Tom Wada)
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Review of cosine and sine
θ r x y HO# /April/21 System Arch (Fire Tom Wada)
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System Arch 2006 (Fire Tom Wada)
Period T0 of sinusoids HO# /April/21 System Arch (Fire Tom Wada)
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QUIZ – what is the equation?-
HO# /April/21 System Arch (Fire Tom Wada)
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System Arch 2006 (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) HO# /April/21 System Arch (Fire Tom Wada)
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Cartesian Form and Polar Form
z=(x, y) y y r θ x x HO# /April/21 System Arch (Fire Tom Wada)
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System Arch 2006 (Fire Tom Wada)
Transform Polar to Cartesian Cartesian to Polar HO# /April/21 System Arch (Fire Tom Wada)
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System Arch 2006 (Fire Tom Wada)
Euler’s law r=1 θ HO# /April/21 System Arch (Fire Tom Wada)
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Complex exponential signal (Rotation Function)
Amplitude = A Θ=ω0t + Φ Φ: phase-shift ω0 : rotation speed + : counterclockwise - : clockwise A -A ω0 radian per second HO# /April/21 System Arch (Fire Tom Wada)
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System Arch 2006 (Fire Tom Wada)
The complex exponential signal is another representation of cosine signal Re means a Projection to X-axis HO# /April/21 System Arch (Fire Tom Wada)
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System Arch 2006 (Fire Tom Wada)
Inverse Euler’s law X* is conjugate of X. HO# /April/21 System Arch (Fire Tom Wada)
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System Arch 2006 (Fire Tom Wada)
Inverse Euler’s law (2) Real signal is the combination of + and - frequency complex exponential signal. HO# /April/21 System Arch (Fire Tom Wada)
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Complex amplitude (Phasor)
X is called as Complex Amplitude or Phasor. HO# /April/21 System Arch (Fire Tom Wada)
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Complex amplitude (Phasor) -2-
1 -1 ω0 radian per second A -A ω0 radian per second X shows the t=0 position. HO# /April/21 System Arch (Fire Tom Wada)
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2 different frequency composition
2 apart frequencies 2 close frequencies Beat HO# /April/21 System Arch (Fire Tom Wada)
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System Arch 2006 (Fire Tom Wada)
Sum of N cosine waves This is sum of 2N+1 different frequency signals. HO# /April/21 System Arch (Fire Tom Wada)
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System Arch 2006 (Fire Tom Wada)
What is Spectrum Spectrum is the set of phasors for each frequencies. HO# /April/21 System Arch (Fire Tom Wada)
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System Arch 2006 (Fire Tom Wada)
Spectrum plot N=3 frequency f (Hz) -f3 -f2 -f1 f1 f2 f3 The line length is proportional to |phasor|. HO# /April/21 System Arch (Fire Tom Wada)
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System Arch 2006 (Fire Tom Wada)
Example frequency f (Hz) -250 -100 100 250 HO# /April/21 System Arch (Fire Tom Wada)
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System Arch 2006 (Fire Tom Wada)
HW1-1 2.30 HO# /April/21 System Arch (Fire Tom Wada)
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System Arch 2006 (Fire Tom Wada)
HW1-2 2.68 HO# /April/21 System Arch (Fire Tom Wada)
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System Arch 2006 (Fire Tom Wada)
HW1-3 3.9 HO# /April/21 System Arch (Fire Tom Wada)
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System Arch 2006 (Fire Tom Wada)
References DSP First – A Multimedia Approach HO# /April/21 System Arch (Fire Tom Wada)
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