OUTLINE n n n Wave Sinusoidal signal review
- Slides: 30
OUTLINE n n n Wave Sinusoidal signal review Complex exponential signal Complex amplitude (phasor) Spectrum 2011/10/24 System Arch 1
2 D wave POND BOAT person can send message by making water wave. LAND person can understand message by the movement of the float. 2011/10/24 System Arch 2
Even with FOG POND LAND person just observe ONLY the float. 2011/10/24 System Arch 3
Even with NOISE POND If main signal is enough stronger than the interference (noise), LAND person can observe the message. 2011/10/24 System Arch 4
Typical wave “Sinusoids” n n 2011/10/24 System Arch t : time A : amplitude ω0 : radian frequency Φ: phase-shift 5
Review of cosine and sine y r θ x 2011/10/24 System Arch 6
Period T 0 of sinusoids 2011/10/24 System Arch 7
QUIZ – what is the equation? - 2011/10/24 System Arch 8
Review of complex n n Cartesian Form q Complex z=(x, y) q x = Real part of z : q y = Imaginary part of z : q z = x + j y Polar Form q 2011/10/24 Complex z=(r, θ) System Arch 9
Cartesian Form and Polar Form z=(x, y) y y r θ x 2011/10/24 x System Arch 10
Transform n Polar to Cartesian 2011/10/24 n System Arch Cartesian to Polar 11
Euler’s law r=1 θ 2011/10/24 System Arch 12
Complex exponential signal (Rotation Function) n n Amplitude = A Θ＝ω0 t + Φ Φ: phase-shift ω0 : rotation speed q q A + : counterclockwise - : clockwise ω0 radian per second A -A -A 2011/10/24 System Arch 13
The complex exponential signal is another representation of cosine signal n Re means a Projection to X-axis 2011/10/24 System Arch 14
Inverse Euler’s law X* is conjugate of X. 2011/10/24 System Arch 15
Inverse Euler’s law (2) n Real signal is the combination of + and frequency complex exponential signal. 2011/10/24 System Arch 16
Complex amplitude (Phasor) n X is called as Complex Amplitude or Phasor. 2011/10/24 System Arch 17
Complex amplitude (Phasor) -21 ω0 radian per second -1 1 -1 A ω0 radian per second A -A • X shows the t=0 position. -A 2011/10/24 System Arch 18
Why Phasor is useful? 1. If you want to add two waves…. 2. Make the complex exponential signal 2011/10/24 System Arch 19
Why Phasor is useful? (2) 3. Make the Phasor… 4. Add them… 2011/10/24 System Arch 20
Why Phasor is useful? (3) 5. Result 2011/10/24 System Arch 21
2 different frequency composition n 2 apart frequencies n 2 close frequencies Beat n 2011/10/24 System Arch 22
Sum of N cosine waves n This is sum of 2 N+1 different frequency signals. 2011/10/24 System Arch 23
What is Spectrum n Spectrum is the set of phasors for each frequencies. 2011/10/24 System Arch 24
Spectrum plot N=3 -f 2 -f 1 0 f 1 f 2 f 3 frequency f (Hz) The line length is proportional to |phasor|. 2011/10/24 System Arch 25
Example -250 2011/10/24 -100 0 100 System Arch 250 frequency f (Hz) 26
HW 1 -1 2. 30 2011/10/24 System Arch 27
HW 1 -2 2. 68 2011/10/24 System Arch 28
HW 1 -3 3. 9 2011/10/24 System Arch 29
HW submission n Submit your Home Work by report 2011/10/24 System Arch 30
