OUTLINE
Wave
Sinusoidal signal review
Complex exponential signal
Complex amplitude (phasor)
Spectrum
2 D wave
POND
Even with FOG
POND
LAND person just observe ONLY the float.
Even with NOISE
POND
Typical wave “Sinusoids”
t : time
A : amplitude
ω0 : radian frequency
Φ: phase-shift
) cos(
)
(t A t x
Review of cosine and sine
θ r
x y
r x r
y
cos sin
cos
sin r
x
r y
Period T0 of sinusoids
) cos(
) cos(
) cos(
) )
( cos(
) ( )
(
0 0
0 0
0 0
0 0
t T
t
t A
T t
A
t x T
t x
2 )
2 (
2
0 0
0 0
T f
T
0
0 0
1 2
T f T
QUIZ – what is the equation?-
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)
ze x
zm y
1
j
Cartesian Form and Polar Form
z=(x, y) y
ze
zm
θ y
ze
zm
re j
jy x
z
r
Transform
Polar to Cartesian Cartesian to Polar
sin cos r
y
r x
x y y x
r
arctan
2 2
Euler’s law
cos j sin
e
j
ze
zm
r=1 θcos
sin
Complex exponential signal (Rotation Function)
Amplitude = A
Θ ＝ ω0t + Φ
Φ: phase-shift
ω0 : rotation speed
+ : counterclockwise
- : clockwise
) sin(
) cos(
)
~(
0 0
) ( 0
t jA
t A
Ae t
x j t
z
e
z
m
-A A
A ω0 radian per second
The complex exponential signal is another representation of
cosine signal
Re means a Projection to X-axis
cos( ))
(t e Ae (0 ) A 0t
x j t
Inverse Euler’s law
~( )
2
)
~ ( )
~(
) 2 cos(
cos 2
*
) (
) (
0
0 0
t x t e
x t
x
e A e
t A
e e
t j
t j
j j
X* is conjugate of X.
Inverse Euler’s law (2)
Real signal is the combination of + and - frequency complex exponential signal.
j e e
e e
j j
j j
sin 2 cos 2
)
cos(
) (
)
(0 0
t A e j t e j t A
Complex amplitude (Phasor)
X is called as Complex Amplitude or Phasor.
t j j
t j j
t j
t j
Xe t
x
Ae X
e Ae
Ae t
x
t A
Ae e
t x
0
0 0
0
)
~(
)
~(
) cos(
) (
) (
0 )
(
Complex amplitude (Phasor)
-2- j
Ae X
A
-1 1
-1
1 ω0 radian per second
t
e
j0 z
e
z
m
-A A
A ω0 radian per second
t
Xe j
t
x ( ) 0
~
Why Phasor is useful?
1. If you want to add two waves….
) 180 /
200 )
10 ( 2 cos(
9 . 1 )
(
) 180 /
70 )
10 ( 2 cos(
7 . 1 )
(
2 1
t t
x
t t
x
2. Make the complex exponential signal
] 9
. 1 [ )
(
] 7
. 1 [ )
(
) 10 ( 2 180
/ 200 2
) 10 ( 2 180
/ 70 1
t j
j
t j
j
e e
e t
x
e e
e t
x
Why Phasor is useful? (2)
3. Make the Phasor…
6498 .
0 785
. 1 9
. 1
597 .
1 5814
. 0 7
. 1
180 / 200 2
180 / 70 1
j e
X
j e
X
j j
4. Add them…
2 1
3
9476 .
0 204
. 1
) 6498 .
0 785
. 1 (
597 .
1 5814
. 0
j
j j
X X
X
Why Phasor is useful? (3)
5. Result
)) 0394 .
0 )(
10 ( 2 cos(
532 .
1
) 180 /
79 . 141 )
10 ( 2 cos(
532 .
1
] 532
. 1 [ )
(
532 .
1
) 10 ( 2 180
/ 79 . 141 3
180 / 79 . 141 3
t t
e e
e t
x
e X
t j
j j
2 different frequency composition
2 apart frequencies
2 close frequencies
Beat
Sum of N cosine waves
This is sum of 2N+1 different frequency signals.
N k
t f k j
t f k j
N k
t f j k j
k k
N k
k k
k
k k
k k
X e X e
X
e X e X
t x
e A X
t f A
A t
x
1
* 2 2
0
1
2 0
1 0
2 2
) (
) 2
cos(
) (
What is Spectrum
Spectrum is the set of phasors for each frequencies.
, ,
2 , 1 2 ,
, 1 2 ,
, 1 2 ,
, 1 0
, 1 1 1* 1 2 2 2* 2
0 X f X f X f X f
X
Spectrum plot
frequency f (Hz)
N=3
0 f1
-f1 f2
-f2 f3
-f3
X 0
2 X1
2
*
X1
2 X 2
2 X3
2
*
X 2
2
*
X 3
N
k
t f k j
t f
k j k X e k
X e X
t x
1
* 2 2
0 2 2
)
(
The line length is proportional to |phasor|.
Example
t j
j t
j j
t j
j t
j j
e e
e e
e e
e e
t t
t x
) 250 ( 2 2
/ )
250 ( 2 2
/
) 100 ( 2 3
/ )
100 ( 2 3
/
4 4
7 7
10
) 2 / 500
cos(
8 )
3 / 200
cos(
14 10
) (
frequency
10 7e j /3 4e j /2
3
7e j / 2
4e j /
HW1-1
2.30
HW1-2
HW1-3
3.9
HW submission
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