2009/10/19 System Arch 1 OUTLINE Wave Sinusoidal signal review Complex exponential signal Complex amplitude (phasor) What is Spectrum.

2009/10/19 System Arch 1

OUTLINE

Wave Sinusoidal signal review Complex exponential signal Complex amplitude (phasor) What is Spectrum

2009/10/19 System Arch 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

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Even with FOG

POND

LAND person just observe ONLY the float.

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Even with NOISE

POND

If main signal is enough stronger than the interference (noise),

LAND person can observe the message.

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Typical wave “Sinusoids”

t : time A : amplitude ω0 : radian frequency Φ: phase-shift

)cos()( tAtx

2009/10/19 System Arch 6

Review of cosine and sine

θ

r

x

y

r

xr

y

cos

sin

cos

sin

rx

ry

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Period T0 of sinusoids

)cos()cos(

)cos())(cos(

)()(

0000

000

0

tTt

tATtA

txTtx

2)2(

2

00

00

Tf

T

00

00

1

2

fT

T

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QUIZ – what is the equation?-

2009/10/19 System Arch 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)

zex

zmy

1j

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Cartesian Form and Polar Form

z=(x, y)

x

y

ze

zm

θx

y

ze

zmjrejyxz

r

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Transform

Polar to Cartesian Cartesian to Polar

sin

cos

ry

rx

x

y

yxr

arctan

22

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Euler’s law

sincos je j

ze

zmr=1

θcos

sin

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Complex exponential signal (Rotation Function)

Amplitude = A Θ ＝ ω0t + Φ Φ: phase-shift ω0 : rotation speed

+ : counterclockwise - : clockwise

)sin()cos(

)(~

00

)( 0

tjAtA

Aetx tj

ze

zm

A-A

-A

A ω0 radian per second

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The complex exponential signal is another representation of cosine signal

Re means a Projection to X-axis

)cos()( 0)( 0 tAAeetx tj

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Inverse Euler’s law

)(~2

)(~)(~2

)cos(

2cos

*

)()(

0

00

txetxtx

eeAtA

ee

tjtj

jj

X* is conjugate of X.

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Inverse Euler’s law (2)

Real signal is the combination of + and - frequency complex exponential signal.

j

ee

ee

jj

jj

2sin

2cos

2)cos(

)()(

0

00

tjtj ee

AtA

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Complex amplitude (Phasor)

X is called as Complex Amplitude or Phasor.

tj

j

tjjtj

tj

Xetx

AeX

eAeAetx

tAAeetx

0

00

0

)(~

)(~)cos()(

)(

0)(

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Complex amplitude (Phasor) -2- jAeX

A

1-1

-1

1ω0 radian per secondtje 0

ze

zm

A-A

-A

A ω0 radian per second

tjXetx 0)(~

• X shows the t=0 position.

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Why Phasor is useful?

1. If you want to add two waves….

)180/200)10(2cos(9.1)(

)180/70)10(2cos(7.1)(

2

1

ttx

ttx

2. Make the complex exponential signal

]9.1[)(

]7.1[)()10(2180/200

2

)10(2180/701

tjj

tjj

eeetx

eeetx

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Why Phasor is useful? (2)3. Make the Phasor…

6498.0785.19.1

597.15814.07.1180/200

2

180/701

jeX

jeXj

j

4. Add them…

180/79.141

213

532.1

9476.0204.1

)6498.0785.1(597.15814.0

je

j

jj

XXX

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Why Phasor is useful? (3)5. Result

))0394.0)(10(2cos(532.1

)180/79.141)10(2cos(532.1

]532.1[)(

532.1)10(2180/79.141

3

180/79.1413

t

t

eeetx

eXtjj

j

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2 different frequency composition

2 apart frequencies

2 close frequencies Beat

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Sum of N cosine waves

This is sum of 2N+1 different frequency signals.

N

k

tfjktfjk

N

k

tfjk

jkk

N

kkkk

kk

k

k

eX

eX

X

eXeXtx

eAX

tfAAtx

1

2*

20

1

20

10

22

)(

)2cos()(

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What is Spectrum

Spectrum is the set of phasors for each frequencies.

,,

2

1,,

2

1,,

2

1,,

2

1,0, 2

*2221

*1110 fXfXfXfXX

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Spectrum plot

frequency f (Hz)

N=3

0 f1-f1 f2-f2 f3-f3

0X21X

2

*1X

22X

23X

2

*2X

2

*3X

N

k

tfjktfjk kk eX

eX

Xtx1

2*

20 22

)(

The line length is proportional to |phasor|.

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Example

tjjtjj

tjjtjj

eeee

eeee

tttx

)250(22/)250(22/

)100(23/)100(23/

44

7710

)2/500cos(8)3/200cos(1410)(

frequency f (Hz)

0 100-100 250-250

103/7 je

2/4 je3/7 je2/4 je

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HW1-1

2.30

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HW1-2

2.68

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HW1-3

3.9

HW submission

Submit your Home Work to Ikenoya-san [email protected]

Subject: SysArch09-HW1 Show your Student ID and Your Name

2009/10/19 System Arch 30

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References

1. DSP First – A Multimedia Approach