2008/10/23 System Arch 2008 (Fire Tom Wada) 1 OUTLINE Wave Sinusoidal signal review Complex exponential signal Complex amplitude (phasor) What is Spectrum

2008/10/23 System Arch 2008 (Fire Tom Wada) 2 2 D wave BOAT person can send message by making water wave. LAND person can understand message by the movement of the float. POND

2008/10/23 System Arch 2008 (Fire Tom Wada) 3 Even with FOG POND LAND person just observe ONLY the float.

2008/10/23 System Arch 2008 (Fire Tom Wada) 4 Even with NOISE POND If main signal is enough stronger than the interference (noise), LAND person can observe the message.

2008/10/23 System Arch 2008 (Fire Tom Wada) 5 Typical wave “ Sinusoids ” t : time A : amplitude ω 0 : radian frequency Φ: phase-shift

2008/10/23 System Arch 2008 (Fire Tom Wada) 6 Review of cosine and sine θ r x y

2008/10/23 System Arch 2008 (Fire Tom Wada) 7 Period T 0 of sinusoids

2008/10/23 System Arch 2008 (Fire Tom Wada) 8 QUIZ – what is the equation?-

2008/10/23 System Arch 2008 (Fire Tom Wada) 9 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)

2008/10/23 System Arch 2008 (Fire Tom Wada) 10 Cartesian Form and Polar Form z=(x, y) x y θ x y r

2008/10/23 System Arch 2008 (Fire Tom Wada) 11 Transform Polar to Cartesian Cartesian to Polar

2008/10/23 System Arch 2008 (Fire Tom Wada) 12 Euler ’ s law r=1 θ

2008/10/23 System Arch 2008 (Fire Tom Wada) 13 Complex exponential signal (Rotation Function) Amplitude = A Θ ＝ ω 0 t + Φ Φ: phase-shift ω 0 : rotation speed  + : counterclockwise  - : clockwise A -A Aω 0 radian per second

2008/10/23 System Arch 2008 (Fire Tom Wada) 14 The complex exponential signal is another representation of cosine signal Re means a Projection to X-axis

2008/10/23 System Arch 2008 (Fire Tom Wada) 15 Inverse Euler ’ s law X* is conjugate of X.

2008/10/23 System Arch 2008 (Fire Tom Wada) 16 Inverse Euler ’ s law (2) Real signal is the combination of + and - frequency complex exponential signal.

2008/10/23 System Arch 2008 (Fire Tom Wada) 17 Complex amplitude (Phasor) X is called as Complex Amplitude or Phasor.

2008/10/23 System Arch 2008 (Fire Tom Wada) 18 Complex amplitude (Phasor) ω 0 radian per second A -A Aω 0 radian per second X shows the t=0 position.

2008/10/23 System Arch 2008 (Fire Tom Wada) 19 Why Phasor is useful? 1. If you want to add two waves…. 2. Make the complex exponential signal

2008/10/23 System Arch 2008 (Fire Tom Wada) 20 Why Phasor is useful? (2) 3. Make the Phasor… 4. Add them…

2008/10/23 System Arch 2008 (Fire Tom Wada) 21 Why Phasor is useful? (3) 5. Result

2008/10/23 System Arch 2008 (Fire Tom Wada) 22 2 different frequency composition 2 apart frequencies 2 close frequencies Beat

2008/10/23 System Arch 2008 (Fire Tom Wada) 23 Sum of N cosine waves This is sum of 2N+1 different frequency signals.

2008/10/23 System Arch 2008 (Fire Tom Wada) 24 What is Spectrum Spectrum is the set of phasors for each frequencies.

2008/10/23 System Arch 2008 (Fire Tom Wada) 25 Spectrum plot frequency f (Hz) N=3 0f1-f1f2-f2f3-f3 The line length is proportional to |phasor|.

2008/10/23 System Arch 2008 (Fire Tom Wada) 26 Example frequency f (Hz)

2008/10/23 System Arch 2008 (Fire Tom Wada) 27 HW

2008/10/23 System Arch 2008 (Fire Tom Wada) 28 HW

2008/10/23 System Arch 2008 (Fire Tom Wada) 29 HW

HW submission Submit your Home Work to Nohara-san Subject: SysArch08-HW1 Show your Student ID and Your Name 2008/10/23 System Arch 2008 (Fire Tom Wada) 30

2008/10/23 System Arch 2008 (Fire Tom Wada) 31 References 1. DSP First – A Multimedia Approach