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 2007 (Fire Tom Wada)
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 2007 (Fire Tom Wada)
System Arch 2007 (Fire Tom Wada) Even with FOG POND LAND person just observe ONLY the float. 2019/4/23 System Arch 2007 (Fire Tom Wada)
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 2007 (Fire Tom Wada)
Typical wave “Sinusoids” t : time A : amplitude ω0 : radian frequency Φ: phase-shift 2019/4/23 System Arch 2007 (Fire Tom Wada)
Review of cosine and sine θ r x y 2019/4/23 System Arch 2007 (Fire Tom Wada)
System Arch 2007 (Fire Tom Wada) Period T0 of sinusoids 2019/4/23 System Arch 2007 (Fire Tom Wada)
QUIZ – what is the equation?- 2019/4/23 System Arch 2007 (Fire Tom Wada)
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 2007 (Fire Tom Wada)
Cartesian Form and Polar Form z=(x, y) y y r θ x x 2019/4/23 System Arch 2007 (Fire Tom Wada)
System Arch 2007 (Fire Tom Wada) Transform Polar to Cartesian Cartesian to Polar 2019/4/23 System Arch 2007 (Fire Tom Wada)
System Arch 2007 (Fire Tom Wada) Euler’s law r=1 θ 2019/4/23 System Arch 2007 (Fire Tom Wada)
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 2007 (Fire Tom Wada)
System Arch 2007 (Fire Tom Wada) The complex exponential signal is another representation of cosine signal Re means a Projection to X-axis 2019/4/23 System Arch 2007 (Fire Tom Wada)
System Arch 2007 (Fire Tom Wada) Inverse Euler’s law X* is conjugate of X. 2019/4/23 System Arch 2007 (Fire Tom Wada)
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 2007 (Fire Tom Wada)
Complex amplitude (Phasor) X is called as Complex Amplitude or Phasor. 2019/4/23 System Arch 2007 (Fire Tom Wada)
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 2007 (Fire Tom Wada)
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 2007 (Fire Tom Wada)
System Arch 2007 (Fire Tom Wada) Why Phasor is useful? (2) Make the Phasor… Add them… 2019/4/23 System Arch 2007 (Fire Tom Wada)
System Arch 2007 (Fire Tom Wada) Why Phasor is useful? (3) Result 2019/4/23 System Arch 2007 (Fire Tom Wada)
2 different frequency composition 2 apart frequencies 2 close frequencies Beat 2019/4/23 System Arch 2007 (Fire Tom Wada)
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 2007 (Fire Tom Wada)
System Arch 2007 (Fire Tom Wada) What is Spectrum Spectrum is the set of phasors for each frequencies. 2019/4/23 System Arch 2007 (Fire Tom Wada)
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 2007 (Fire Tom Wada)
System Arch 2007 (Fire Tom Wada) Example frequency f (Hz) -250 -100 100 250 2019/4/23 System Arch 2007 (Fire Tom Wada)
System Arch 2007 (Fire Tom Wada) HW1-1 2.30 2019/4/23 System Arch 2007 (Fire Tom Wada)
System Arch 2007 (Fire Tom Wada) HW1-2 2.68 2019/4/23 System Arch 2007 (Fire Tom Wada)
System Arch 2007 (Fire Tom Wada) HW1-3 3.9 2019/4/23 System Arch 2007 (Fire Tom Wada)
System Arch 2007 (Fire Tom Wada) References DSP First – A Multimedia Approach 2019/4/23 System Arch 2007 (Fire Tom Wada)