2009/10/19 system arch 1 outline wave sinusoidal signal review complex exponential signal complex...
TRANSCRIPT
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
2009/10/19 System Arch 3
Even with FOG
POND
LAND person just observe ONLY the float.
2009/10/19 System Arch 4
Even with NOISE
POND
If main signal is enough stronger than the interference (noise),
LAND person can observe the message.
2009/10/19 System Arch 5
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
2009/10/19 System Arch 7
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
2009/10/19 System Arch 8
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
2009/10/19 System Arch 10
Cartesian Form and Polar Form
z=(x, y)
x
y
ze
zm
θx
y
ze
zmjrejyxz
r
2009/10/19 System Arch 11
Transform
Polar to Cartesian Cartesian to Polar
sin
cos
ry
rx
x
y
yxr
arctan
22
2009/10/19 System Arch 12
Euler’s law
sincos je j
ze
zmr=1
θcos
sin
2009/10/19 System Arch 13
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
2009/10/19 System Arch 14
The complex exponential signal is another representation of cosine signal
Re means a Projection to X-axis
)cos()( 0)( 0 tAAeetx tj
2009/10/19 System Arch 15
Inverse Euler’s law
)(~2
)(~)(~2
)cos(
2cos
*
)()(
0
00
txetxtx
eeAtA
ee
tjtj
jj
X* is conjugate of X.
2009/10/19 System Arch 16
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
2009/10/19 System Arch 17
Complex amplitude (Phasor)
X is called as Complex Amplitude or Phasor.
tj
j
tjjtj
tj
Xetx
AeX
eAeAetx
tAAeetx
0
00
0
)(~
)(~)cos()(
)(
0)(
2009/10/19 System Arch 18
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.
2009/10/19 System Arch 19
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
2009/10/19 System Arch 20
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
2009/10/19 System Arch 21
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
2009/10/19 System Arch 22
2 different frequency composition
2 apart frequencies
2 close frequencies Beat
2009/10/19 System Arch 23
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()(
2009/10/19 System Arch 24
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
2009/10/19 System Arch 25
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|.
2009/10/19 System Arch 26
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
2009/10/19 System Arch 27
HW1-1
2.30
2009/10/19 System Arch 28
HW1-2
2.68
2009/10/19 System Arch 29
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
2009/10/19 System Arch 31
References
1. DSP First – A Multimedia Approach