OUTLINE Wave Sinusoidal signal review Complex exponential signal Complex amplitude (phasor) Spectrum 2012/10/29 System Arch

2 D wave POND BOAT person can send message by making water wave. LAND person can understand message by the movement of the float. 2012/10/29 System Arch

Even with FOG POND LAND person just observe ONLY the float. 2012/10/29 System Arch

Even with NOISE POND If main signal is enough stronger than the interference (noise), LAND person can observe the message. 2012/10/29 System Arch

Typical wave “Sinusoids” t : time A : amplitude ω0 : radian frequency Φ: phase-shift 2012/10/29 System Arch

Review of cosine and sine θ r x y 2012/10/29 System Arch

Period T0 of sinusoids 2012/10/29 System Arch

QUIZ – what is the equation?- 2012/10/29 System Arch

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) 2012/10/29 System Arch

Cartesian Form and Polar Form z=(x, y) y y r θ x x 2012/10/29 System Arch

Transform Polar to Cartesian Cartesian to Polar 2012/10/29 System Arch

Euler’s law r=1 θ 2012/10/29 System Arch

Complex exponential signal (Rotation Function) Amplitude = A Θ＝ω0t + Φ Φ: phase-shift ω0 : rotation speed + : counterclockwise - : clockwise A -A ω0 radian per second 2012/10/29 System Arch

The complex exponential signal is another representation of cosine signal Re means a Projection to X-axis 2012/10/29 System Arch

Inverse Euler’s law X* is conjugate of X. 2012/10/29 System Arch

Inverse Euler’s law (2) Real signal is the combination of + and - frequency complex exponential signal. 2012/10/29 System Arch

Complex amplitude (Phasor) X is called as Complex Amplitude or Phasor. 2012/10/29 System Arch

Complex amplitude (Phasor) -2- 1 -1 ω0 radian per second A -A ω0 radian per second X shows the t=0 position. 2012/10/29 System Arch

Why Phasor is useful? If you want to add two waves…. Make the complex exponential signal 2012/10/29 System Arch

Why Phasor is useful? (2) Make the Phasor… Add them… 2012/10/29 System Arch

Why Phasor is useful? (3) Result 2012/10/29 System Arch

2 different frequency composition 2 apart frequencies 2 close frequencies Beat 2012/10/29 System Arch

Sum of N cosine waves This is sum of 2N+1 different frequency signals. 2012/10/29 System Arch

What is Spectrum Spectrum is the set of phasors for each frequencies. 2012/10/29 System Arch

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|. 2012/10/29 System Arch

Example frequency f (Hz) -250 -100 100 250 2012/10/29 System Arch

HW1-1 2.30 2012/10/29 System Arch

HW1-2 2.68 2012/10/29 System Arch

HW1-3 3.9 2012/10/29 System Arch

HW submission Submit your Home Work by report 2012/10/29 System Arch