OUTLINE n n n Wave Sinusoidal signal review

OUTLINE n n n Wave Sinusoidal signal review Complex exponential signal Complex amplitude (phasor) What is Spectrum 11/22/2020 System Arch 2007 (Fire Tom Wada) 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. 11/22/2020 System Arch 2007 (Fire Tom Wada) 2

Even with FOG POND LAND person just observe ONLY the float. 11/22/2020 System Arch 2007 (Fire Tom Wada) 3

Even with NOISE POND If main signal is enough stronger than the interference (noise), LAND person can observe the message. 11/22/2020 System Arch 2007 (Fire Tom Wada) 4

Typical wave “Sinusoids” n n 11/22/2020 t : time A : amplitude ω0 : radian frequency Φ: phase-shift System Arch 2007 (Fire Tom Wada) 5

Review of cosine and sine y r θ x 11/22/2020 System Arch 2007 (Fire Tom Wada) 6

Period T 0 of sinusoids 11/22/2020 System Arch 2007 (Fire Tom Wada) 7

QUIZ – what is the equation? - 11/22/2020 System Arch 2007 (Fire Tom Wada) 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 11/22/2020 Complex z=(r, θ) System Arch 2007 (Fire Tom Wada) 9

Cartesian Form and Polar Form z=(x, y) y y r θ x 11/22/2020 x System Arch 2007 (Fire Tom Wada) 10

Transform n Polar to Cartesian 11/22/2020 n Cartesian to Polar System Arch 2007 (Fire Tom Wada) 11

Euler’s law r=1 θ 11/22/2020 System Arch 2007 (Fire Tom Wada) 12

Complex exponential signal (Rotation Function) n n Amplitude = A Θ＝ω0 t + Φ Φ: phase-shift ω0 : rotation speed q q + : counterclockwise - : clockwise A ω0 radian per second A -A -A 11/22/2020 System Arch 2007 (Fire Tom Wada) 13

The complex exponential signal is another representation of cosine signal n Re means a Projection to X-axis 11/22/2020 System Arch 2007 (Fire Tom Wada) 14

Inverse Euler’s law X* is conjugate of X. 11/22/2020 System Arch 2007 (Fire Tom Wada) 15

Inverse Euler’s law (2) n Real signal is the combination of + and frequency complex exponential signal. 11/22/2020 System Arch 2007 (Fire Tom Wada) 16

Complex amplitude (Phasor) n X is called as Complex Amplitude or Phasor. 11/22/2020 System Arch 2007 (Fire Tom Wada) 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 11/22/2020 System Arch 2007 (Fire Tom Wada) 18

Why Phasor is useful? 1. If you want to add two waves…. 2. Make the complex exponential signal 11/22/2020 System Arch 2007 (Fire Tom Wada) 19

Why Phasor is useful? (2) 3. Make the Phasor… 4. Add them… 11/22/2020 System Arch 2007 (Fire Tom Wada) 20

Why Phasor is useful? (3) 5. Result 11/22/2020 System Arch 2007 (Fire Tom Wada) 21

2 different frequency composition n 2 apart frequencies n 2 close frequencies Beat n 11/22/2020 System Arch 2007 (Fire Tom Wada) 22

Sum of N cosine waves n This is sum of 2 N+1 different frequency signals. 11/22/2020 System Arch 2007 (Fire Tom Wada) 23

What is Spectrum n Spectrum is the set of phasors for each frequencies. 11/22/2020 System Arch 2007 (Fire Tom Wada) 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|. 11/22/2020 System Arch 2007 (Fire Tom Wada) 25

Example -250 11/22/2020 -100 0 100 System Arch 2007 (Fire Tom Wada) 250 frequency f (Hz) 26

HW 1 -1 2. 30 11/22/2020 System Arch 2007 (Fire Tom Wada) 27

HW 1 -2 2. 68 11/22/2020 System Arch 2007 (Fire Tom Wada) 28

HW 1 -3 3. 9 11/22/2020 System Arch 2007 (Fire Tom Wada) 29

References 1. DSP First – A Multimedia Approach 11/22/2020 System Arch 2007 (Fire Tom Wada) 30