View
0
Download
0
Category
Preview:
Citation preview
1
Principles of CommunicationsECS 332
Asst. Prof. Dr. Prapun Suksompongprapun@siit.tu.ac.th
4. Amplitude Modulation
Office Hours: BKD, 4th floor of Sirindhralai building
Monday 9:30-10:30Monday 14:00-16:00Thursday 16:00-17:00
DSB-SC
2
× ×Channel
2 cos 2 cf t
y
2 cos 2 cf t
vLPF
Modulator Demodulator
Message(modulating signal)
22
2 cos 2 2 cos 2c cx t
v t
m t f t f t
LPF m tKey equation:
In the time domain…
3
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-2
-1
0
1
2
3
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-4
-2
0
2
4
2
2cos 2
2
2cos 2
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-5
0
5
Seconds
Note the oscillation at twice the carrier frequency
In the time domain…
4
2
2cos 2
2
2cos 2
When the sampling rate is not fast enough,…
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-5
0
5
Seconds
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-4
-2
0
2
4
Seconds
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-2
-1
0
1
2
3
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-3
-2
-1
0
1
2
3
4
5
The problem with sampling rate
5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-5
0
5
Seconds
This is the plot of when we don’t connect the dots
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-3
-2
-1
0
1
2
3
4
5
The problem with sampling rate
6 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-5
0
5
Seconds
DSB-SC
7
0 5 10 15 20 25-1
-0.5
0
0.5
1
Seconds
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
x 104
0
0.05
0.1
0.15
0.2
Frequency [Hz]
Mag
nitu
de
0 5 10 15 20 25-2
-1
0
1
2
Seconds
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
x 104
0
0.05
0.1
0.15
0.2
Frequency [Hz]
Mag
nitu
de0 5 10 15 20 25
-2
-1
0
1
2
Seconds
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
x 104
0
0.05
0.1
0.15
0.2
Frequency [Hz]
Mag
nitu
de
0 5 10 15 20 25-2
-1
0
1
2
Seconds
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
x 104
0
0.05
0.1
0.15
0.2
Frequency [Hz]
Mag
nitu
de
/ 2
/2
[Demo_DSBSC_Sound_ReadWAV.m]
DSB-SC (Zoomed in time)
8
1 1.0005 1.001 1.0015 1.002 1.0025 1.003 1.0035 1.004 1.0045 1.005-1
-0.5
0
0.5
1
Seconds
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
x 104
0
0.05
0.1
0.15
0.2
Frequency [Hz]
Mag
nitu
de
1 1.0005 1.001 1.0015 1.002 1.0025 1.003 1.0035 1.004 1.0045 1.005-1
-0.5
0
0.5
1
Seconds
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
x 104
0
0.05
0.1
0.15
0.2
Frequency [Hz]
Mag
nitu
de1 1.0005 1.001 1.0015 1.002 1.0025 1.003 1.0035 1.004 1.0045 1.005
-1
-0.5
0
0.5
1
Seconds
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
x 104
0
0.05
0.1
0.15
0.2
Frequency [Hz]
Mag
nitu
de
Note how the baseband signal becomes the envelope of
the modulated signal x .
Note the delay caused by the LPF.
1 1.0005 1.001 1.0015 1.002 1.0025 1.003 1.0035 1.004 1.0045 1.005-2
-1
0
1
2
Seconds
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
x 104
0
0.05
0.1
0.15
0.2
Frequency [Hz]
Mag
nitu
de
Fourier Series: Ex 1
9
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
t
ECS332_4_Amplitude_Modulation_Fourier_Ex1.fig
Fourier Series: Ex 1
10
-1-0.8
-0.6-0.4
-0.20
0.20.4
0.60.8
1 -0.8-0.6
-0.4-0.2
00.2
0.40.6
0.81
1.2
0
2
4
6
8
10
12
14
16
18
20
inde
x
t
Fourier Series: Ex 1 (interactive)
11 ECS332_4_Amplitude_Modulation_Fourier_Ex1.jar [http://www.tomasboril.cz/hobbies_programs_en.html]
Fourier Series: Ex 1
12
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
t
Fourier Series: Ex 2
13
0 2 4 6 8 10 12-1.5
-1
-0.5
0
0.5
1
1.5
t
ECS332_4_Amplitude_Modulation_Fourier_Ex2.fig
Fourier Series: Ex 2
14
0
2
4
6
8
10
12-1.5
-1
-0.5
0
0.5
1
1.5
0
2
4
6
8
10
12
14
16
18
20
inde
x
t
Fourier Series: Ex 2
15
Fourier Series: Ex 2
16 [http://codepen.io/anon/pen/jPGJMK/]
Fourier Series visualization
17 [http://bl.ocks.org/jinroh/7524988]
Fourier Series: Drawing
18
The same technique, but now tracing the whole trajectory and not just the vertical displacement, can be used to draw “anything”.
[https://www.youtube.com/watch?v=QVuU2YCwHjw&t=1m]
Fourier Series: Drawing
19[http://devpost.com/software/draw-anything]
Draw Anything is an iOS app that harnesses the computational power of the Wolfram Programming Cloud to automatically create step-by-step drawing guides.
Fourier Series: Ex 1
20
0 2-
1
1/1/
sincsin 1
sin
sin 2 sinc 2
-3 113 2
2 13
-2 0 0
-1 -11
1 2
2
0 0 1
1 11
1 2
2
2 0 0
3 -11
3 2
2 13
2
2 13
2 15
Fourier Series: Ex 1
21
0
1
1/1/
2
2 13
2 15
2
21
Fourier Series: Ex 1
22
0
1/2
1
1 13
1 15
2
Note that this is the scaled Fourier transform of the restricted (one period) version of your signal.
Fourier Series: Ex 1
23
0
1/2
1
1 13
1 15
2
These “lines” are collectively referred to as the (two-sided) line spectrum of the periodic signal.
Usually, you will get complex numbers and hence the spectrum is represented by two plots: the amplitude (magnitude) and the phase.
Here, we “happen” to have all of the Fourier coeff. being real-valued. So, one plot is OK.
Fourier Series: Ex 1
24
0
1/2
1
1 13
1 15
2
Simply changing them to “arrows”. Collectively, they are now the Fourier transform of your periodic signal.
Effect of Duty Cycle
25
Effect of Duty Cycle
26
Note that it is not always the case that the 2nd
harmonic (along with its muliples) is suppressed.
Duty cycle = 0.070
Effect of Duty Cycle
27
Duty cycle = 0.125
When duty cycle = 1/8, the 8th harmonic (along with its muliples) is suppressed.
Effect of Duty Cycle
28
When duty cycle = 1/5, the 5th harmonic (along with its muliples) is suppressed.
Duty cycle = 0.203
Effect of Duty Cycle
29
When duty cycle = 1/3, the 3rd harmonic (along with its muliples) is suppressed.
Duty cycle = 0.336
Effect of Duty Cycle
30
When duty cycle = 1/2, the 2nd harmonic (along with its muliples) is suppressed.
Duty cycle = 0.500
Square Wave
31
1
44
0
1/2
1
1 13
1 15
2
Fourier series expansion:
: the scaled Fourier
transform of the restricted (one period) version of .
12
1 13
15 ⋯
1 13
15 ⋯
period
Fundamental frequency = 1/T0
Square Wave
32
1
44Fourier series expansion:
These “lines” are collectively referred to as the (two-sided) line spectrum of the periodic signal.
0
1/2
1
1 13
1 15
2
12
1 13
15 ⋯
1 13
15 ⋯
Square Wave
33
1
44Fourier series expansion:12
1 13
15 ⋯
1 13
15 ⋯
Simply changing them to “arrows” (representing the delta functions). Collectively, they are now the Fourier transform of your periodic signal.
0
1/2
1
1 13
1 15
2
Square Wave
34
1
44Fourier series expansion:12
1 13
15 ⋯
1 13
15 ⋯
12
1 13 3
15 5 ⋯
1 1
3 315 5 ⋯
Square Wave
35
1
44Fourier series expansion:12
1 13
15 ⋯
1 13
15 ⋯
12
2cos 2
23 cos 2 3
25 cos 2 5 ⋯
Trigonometric Fourier series expansion: 2cos
Square Wave
36
1
44
Compact expression based on the cosine function:
Trigonometric Fourier series expansion:
1 cos 2 0 1, cos 2 0,0, otherwise.
12
2cos 2
23 cos 2 3
25 cos 2 5 ⋯
Switching Operation
37
44
OFF ON OFF ON OFF ON OFF ON OFF ON OFF
Switching Operation
38
1
44
OFF ON OFF ON OFF ON OFF ON OFF ON OFF
Multiplying a signal by the square-wave is equivalent to switching on (for half a period) and off periodically.
Switching Modulator
39
12
2cos 2
23 cos 2 3
25 cos 2 5 ⋯
12
2cos 2
23 cos 2 3
25 cos 2 5 ⋯
0
2
53
355 3
Set =
Switching Modulator
40
0
2
53
355 3
BPF2
cos 2
12
2cos 2
23 cos 2 3
25 cos 2 5 ⋯
12
2cos 2
23 cos 2 3
25 cos 2 5 ⋯
Switching Demodulator
41
12
2cos 2
23 cos 2 3
25 cos 2 5 ⋯
LPFcos 2
12
2cos 2
23 cos 2 3
25 cos 2 5 ⋯
Switching Demodulator
42
1 2 2 2cos 2 cos 2 3 cos 2 52 3 512
2 cos 2
2 cos 2 332
cos 2
cos 2
cos 2
co cos 55
s 2 2
c c c
c
c
c c
c c
c c
c cc
y t y t y t y t y t
A m t f t
A
f t f t f t
f t
f t
m t f t
A m t f t
A m t
r
f
t
f tt
1 cos 2 2
cos 2 2 cos 2 4
c
cos 212
1
131
5os 2 4 cos 2 6
c c
c
c c c
cc
c
c
f t
f t
A m t f t
A m t
A m t
A m t
f t
f t f t
12
1 1 cos 2 2
cos 2 2 cos 2 4
co
c
s 2 4 cos 2 6
1 13 31 1
5
2
5
osc c
c c
c c
c cc
c
c c
c
f t
f t
A m t f t
A m t A m t
A m t A m t
A m
f t
f t f tt A m t
cos cos12 cos
12 cos
Switching Demodulator
43
cos 2
1 2 2 2cos 2 cos 2 3 cos 2 52 3 512
2 cos 2
2 cos 2 332
cos 2
cos 2
cos 2
co cos 55
s 2 2
c c c
c
c
c c
c c
c c
c cc
y t y t y t y t y t
A m t f t
A
f t f t f t
f t
f t
m t f t
A m t f t
A m t
r
f
t
f tt
Now, recall that cos cos cos cos
Switching Demodulator
44
1 cos 2 2
1 co
12
s 2 cos 2 5
1 cos 2 3 cos 2
cos
57
2
1
3
c
c c
c c
c c
c
c
c
r t
f t
f
y t A m t f t
A m t
A m t
A m t
t f t
f t f t
cos 2
Switching Demodulator
45
cos 2 2
1 1cos 2 2 cos 2 4
1 1co
12
1 1
3 3
5s 2 4 cos 2 6
5
cos 2c c
c c
c c
c c
c
c c
c c
r t
f t
f
y t A m t f t
A m t A m t
A m t A mt f t
f t
t
A m t A m tt f
cos 2
Switching Demodulator
46
LPFcos 2
cos 2 2
cos 2 2 cos 2 4
cos
12
1 1
1 13
2 4
31 1 cos 2 6
5 5
cos 2c c
c c
c c
c c
c
c c
c c
r t
f t
f
y t A m t f t
A m t A m t
A m t A mt f t
f t
t
A m t A m tt f
Part A
47
outL
+vin-+vin-
240 V
[Slides from basic EE lab]
Part A: Half-Wave Rectifier (HWR)
48
A rectifier is an electrical device that converts alternating current (AC) to direct current (DC).
240 V
[Slides from basic EE lab]
Part A: Half-Wave Rectifier (HWR)
49
=
[Slides from basic EE lab]
Part A: Full-Wave Rectifier (FWR)
50
T1
220 V50 Hz
A
B
C
D
+
_Vout
D11N4001
D21N4001
RL10 k
S1+vin-+vin-
S2
240 V
[Slides from basic EE lab]
Part A: Full-Wave Rectifier (FWR)
51
=
[Slides from basic EE lab]
T1
240 V50 Hz +
_
Vout
D11N4001
D21N4001
A
B
C R110k
C1100 F50 V
+
_
D
Part B: Filter Capacitor
52
240 V
ripple waveform
[Slides from basic EE lab]
53
[Slides from basic EE lab]
Problem with the angle
54
Not as easy as it looks
Problem with the angle
55
atan: Inverse tangent (arctangent)
56
atan
[rad
ians
]
Return values in the interval [-/2,/2]. not (-,]
>> atan(1/1)*180/pians =
45>> atan((-1)/(-1))*180/pians =
45>> atan(-1/1)*180/pians =
-45>> atan(1/-1)*180/pians =
-45
Want this to be -135.
Want this to be 135.
x
y
atan2: Four-quadrant inverse tangent
57
>> atan(1/1)*180/pians =
45>> atan((-1)/(-1))*180/pians =
45>> atan(-1/1)*180/pians =
-45>> atan(1/-1)*180/pians =
-45
Want this to be -135.
Want this to be 135.
x
y
>> atan2(1,1)*180/pians =
45>> atan2(-1,-1)*180/pians =
-135>> atan2(-1,1)*180/pians =
-45>> atan2(1,-1)*180/pians =
135
atan2(y,x) returns values in the interval (-,].
atan2: Four-quadrant inverse tangent
58
atan2 ,
arctan , 0,
arctan , 0 , 0,
arctan , 0 , 0,
2 , 0 , 0,
2 , 0 , 0,
undefined, 0 , 0,
Supplementary Reference
59
Modem Theory: An Introduction to Telecommunications
By Richard E. Blahut Date Published: December 2009 ISBN: 9780521780148 http://www.cambridge.org/us/ac
ademic/subjects/engineering/communications-and-signal-processing/modem-theory-introduction-telecommunications
https://books.google.co.th/books?id=ApmsJAvnMc0C
Richard Blahut
60
Former chair of the Electrical and Computer Engineering Department at the University of Illinois at Urbana-Champaign
Best known for Blahut–Arimotoalgorithm
Claude E. Shannon Award
61
Claude E. Shannon (1972)
David S. Slepian (1974)
Robert M. Fano (1976)
Peter Elias (1977)
Mark S. Pinsker (1978)
Jacob Wolfowitz (1979)
W. Wesley Peterson (1981)
Irving S. Reed (1982)
Robert G. Gallager (1983)
Solomon W. Golomb (1985)
William L. Root (1986)
James L. Massey (1988)
Thomas M. Cover (1990)
Andrew J. Viterbi (1991)
Elwyn R. Berlekamp (1993)
Aaron D. Wyner (1994)
G. David Forney, Jr. (1995)
Imre Csiszár (1996)
Jacob Ziv (1997)
Neil J. A. Sloane (1998)
Tadao Kasami (1999)
Thomas Kailath (2000)
Jack KeilWolf (2001)
Toby Berger (2002)
Lloyd R. Welch (2003)
Robert J. McEliece (2004)
Richard Blahut (2005)
Rudolf Ahlswede (2006)
Sergio Verdu (2007)
Robert M. Gray (2008)
Jorma Rissanen (2009)
Te Sun Han (2010)
Shlomo Shamai (Shitz) (2011)
Abbas El Gamal (2012)
Katalin Marton (2013)
János Körner (2014)
Arthur Robert Calderbank (2015)
Berger plaque
62
Recommended