6.003: Signals and Systems
Modulation
December 6, 2011
1
Communications Systems
Signals are not always well matched to the media through which we
wish to transmit them.
signal
applications
audio
telephone, radio, phonograph, CD, cell phone, MP3
video
television, cinema, HDTV, DVD
internet
coax, twisted pair, cable TV, DSL, optical fiber, E/M
Modulation can improve match based on frequency.
2
Amplitude Modulation
Amplitude modulation can be used to match audio frequencies to
radio frequencies. It allows parallel transmission of multiple channels.
z1(t)
x1(t)
cos w1t
z2(t)
x2(t)
z(t)
cos wct
cos w2t
z3(t)
x3(t)
cos w3t
3
LPF
y(t)
Superheterodyne Receiver
Edwin Howard Armstrong invented the superheterodyne receiver,
which made broadcast AM practical.
Edwin Howard Armstrong also invented and
patented the “regenerative” (positive feedback)
circuit for amplifying radio signals (while he was
a junior at Columbia University). He also in­
vented wide-band FM.
4
Amplitude, Phase, and Frequency Modulation
There are many ways to embed a “message” in a carrier.
Amplitude Modulation (AM) + carrier: y1 (t) = x(t) + C cos(ωc t)
Phase Modulation (PM):
y2 (t) = cos(ωc t + kx(t))
Frequency Modulation (FM):
y3 (t) = cos ωc t + k −∞ x(τ )dτ
t
PM: signal modulates instantaneous phase of the carrier.
y2 (t) = cos(ωc t + kx(t))
FM: signal modulates instantaneous frequency of carrier.
t
y3 (t) = cos ωc t + k
x(τ )dτ
−∞
'
v
"
φ(t)
d
ωi (t) = ωc + φ(t) = ωc + kx(t)
dt
5
Frequency Modulation
Compare AM to FM for x(t) = cos(ωm t).
AM: y1 (t) = x(t) + C cos(ωc t) = (cos(ωm t) + 1.1) cos(ωc t)
t
Rt
FM: y3 (t) = cos ωc t + k −∞ x(τ )dτ = cos(ωc t + ωkm sin(ωm t))
t
Advantages of FM:
• constant power
• no need to transmit carrier (unless DC important)
• bandwidth?
6
Frequency Modulation
Early investigators thought that narrowband FM could have arbitrar­
ily narrow bandwidth, allowing more channels than AM.
Z t
y3 (t) = cos ωc t + k
x(τ )dτ
' −∞v
"
φ(t)
d
ωi (t) = ωc + φ(t) = ωc + kx(t)
dt
Small k → small bandwidth. Right?
7
Frequency Modulation
Early investigators thought that narrowband FM could have arbitrar­
ily narrow bandwidth, allowing more channels than AM. Wrong!
0
Z t
y3 (t) = cos ωc t + k
x(τ )dτ
−∞
0 Z t
0 Z t
= cos(ωc t) × cos k
x(τ )dτ − sin(ωc t) × sin k
x(τ )dτ
−∞
If k → 00 then
Z t
cos k
x(τ )dτ
−∞
0 Z t
sin k
x(τ )dτ
−∞
→1
Z t
→k
x(τ )dτ
0 Z t
y3 (t) ≈ cos(ωc t) − sin(ωc t) × k
x(τ )dτ
−∞
−∞
−∞
Bandwidth of narrowband FM is the same as that of AM!
(integration does not change the highest frequency in the signal)
8
Phase/Frequency Modulation
Find the Fourier transform of a PM/FM signal.
y(t) = cos(ωc t + mx(t)) = cos(ωc t + m sin(ωm t))
= cos(ωc t) cos(m sin(ωm t))) − sin(ωc t) sin(m sin(ωm t)))
, therefore cos(m sin(ωm t)) is periodic in T .
x(t) is periodic in T = ω2π
m
1 sin(ωm t)
1
t
0
−1
cos(1 sin(ωm t))
1
t
0
−1
9
Phase/Frequency Modulation
Find the Fourier transform of a PM/FM signal.
y(t) = cos(ωc t + mx(t)) = cos(ωc t + m sin(ωm t))
= cos(ωc t) cos(m sin(ωm t))) − sin(ωc t) sin(m sin(ωm t)))
, therefore cos(m sin(ωm t)) is periodic in T .
x(t) is periodic in T = ω2π
m
2 sin(ωm t)
2
t
0
−2
cos(2 sin(ωm t))
1
t
0
−1
10
Phase/Frequency Modulation
Find the Fourier transform of a PM/FM signal.
y(t) = cos(ωc t + mx(t)) = cos(ωc t + m sin(ωm t))
= cos(ωc t) cos(m sin(ωm t))) − sin(ωc t) sin(m sin(ωm t)))
, therefore cos(m sin(ωm t)) is periodic in T .
x(t) is periodic in T = ω2π
m
3 sin(ωm t)
3
t
0
−3
cos(3 sin(ωm t))
1
t
0
−1
11
Phase/Frequency Modulation
Find the Fourier transform of a PM/FM signal.
y(t) = cos(ωc t + mx(t)) = cos(ωc t + m sin(ωm t))
= cos(ωc t) cos(m sin(ωm t))) − sin(ωc t) sin(m sin(ωm t)))
, therefore cos(m sin(ωm t)) is periodic in T .
x(t) is periodic in T = ω2π
m
4 sin(ωm t)
4
t
0
−4
cos(4 sin(ωm t))
1
t
0
−1
12
Phase/Frequency Modulation
Find the Fourier transform of a PM/FM signal.
y(t) = cos(ωc t + mx(t)) = cos(ωc t + m sin(ωm t))
= cos(ωc t) cos(m sin(ωm t))) − sin(ωc t) sin(m sin(ωm t)))
, therefore cos(m sin(ωm t)) is periodic in T .
x(t) is periodic in T = ω2π
m
5 sin(ωm t)
5
t
0
−5
cos(5 sin(ωm t))
1
t
0
−1
13
Phase/Frequency Modulation
Find the Fourier transform of a PM/FM signal.
y(t) = cos(ωc t + mx(t)) = cos(ωc t + m sin(ωm t))
= cos(ωc t) cos(m sin(ωm t))) − sin(ωc t) sin(m sin(ωm t)))
, therefore cos(m sin(ωm t)) is periodic in T .
x(t) is periodic in T = ω2π
m
6 sin(ωm t)
6
t
0
−6
cos(6 sin(ωm t))
1
t
0
−1
14
Phase/Frequency Modulation
Find the Fourier transform of a PM/FM signal.
y(t) = cos(ωc t + mx(t)) = cos(ωc t + m sin(ωm t))
= cos(ωc t) cos(m sin(ωm t))) − sin(ωc t) sin(m sin(ωm t)))
, therefore cos(m sin(ωm t)) is periodic in T .
x(t) is periodic in T = ω2π
m
7 sin(ωm t)
7
t
0
−7
cos(7 sin(ωm t))
1
t
0
−1
15
Phase/Frequency Modulation
Find the Fourier transform of a PM/FM signal.
y(t) = cos(ωc t + mx(t)) = cos(ωc t + m sin(ωm t))
= cos(ωc t) cos(m sin(ωm t))) − sin(ωc t) sin(m sin(ωm t)))
, therefore cos(m sin(ωm t)) is periodic in T .
x(t) is periodic in T = ω2π
m
8 sin(ωm t)
8
t
0
−8
cos(8 sin(ωm t))
1
t
0
−1
16
Phase/Frequency Modulation
Find the Fourier transform of a PM/FM signal.
y(t) = cos(ωc t + mx(t)) = cos(ωc t + m sin(ωm t))
= cos(ωc t) cos(m sin(ωm t))) − sin(ωc t) sin(m sin(ωm t)))
, therefore cos(m sin(ωm t)) is periodic in T .
x(t) is periodic in T = ω2π
m
9 sin(ωm t)
9
t
0
−9
cos(9 sin(ωm t))
1
t
0
−1
17
Phase/Frequency Modulation
Find the Fourier transform of a PM/FM signal.
y(t) = cos(ωc t + mx(t)) = cos(ωc t + m sin(ωm t))
= cos(ωc t) cos(m sin(ωm t))) − sin(ωc t) sin(m sin(ωm t)))
, therefore cos(m sin(ωm t)) is periodic in T .
x(t) is periodic in T = ω2π
m
10 sin(ωm t)
10
t
0
−10
cos(10 sin(ωm t))
1
t
0
−1
18
Phase/Frequency Modulation
Find the Fourier transform of a PM/FM signal.
y(t) = cos(ωc t + mx(t)) = cos(ωc t + m sin(ωm t))
= cos(ωc t) cos(m sin(ωm t))) − sin(ωc t) sin(m sin(ωm t)))
, therefore cos(m sin(ωm t)) is periodic in T .
x(t) is periodic in T = ω2π
m
20 sin(ωm t)
20
t
0
−20
cos(20 sin(ωm t))
1
t
0
−1
19
Phase/Frequency Modulation
Find the Fourier transform of a PM/FM signal.
y(t) = cos(ωc t + mx(t)) = cos(ωc t + m sin(ωm t))
= cos(ωc t) cos(m sin(ωm t))) − sin(ωc t) sin(m sin(ωm t)))
, therefore cos(m sin(ωm t)) is periodic in T .
x(t) is periodic in T = ω2π
m
50 sin(ωm t)
50
t
0
−50
cos(50 sin(ωm t))
1
t
0
−1
20
Phase/Frequency Modulation
Find the Fourier transform of a PM/FM signal.
y(t) = cos(ωc t + mx(t)) = cos(ωc t + m sin(ωm t))
= cos(ωc t) cos(m sin(ωm t))) − sin(ωc t) sin(m sin(ωm t)))
, therefore cos(m sin(ωm t)) is periodic in T .
x(t) is periodic in T = ω2π
m
m sin(ωm t)
m
t
0
−m
cos(m sin(ωm t))
1
t
increasing m
0
−1
21
Phase/Frequency Modulation
Find the Fourier transform of a PM/FM signal.
y(t) = cos(ωc t + mx(t)) = cos(ωc t + m sin(ωm t))
= cos(ωc t) cos(m sin(ωm t))) − sin(ωc t) sin(m sin(ωm t)))
x(t) is periodic in T = ω2π
, therefore cos(m sin(ωm t)) is periodic in T .
m
cos(m sin(ωm t))
1
t
0
−1
m=0
|ak |
0
10
20
30
22
40
50
60
k
Phase/Frequency Modulation
Find the Fourier transform of a PM/FM signal.
y(t) = cos(ωc t + mx(t)) = cos(ωc t + m sin(ωm t))
= cos(ωc t) cos(m sin(ωm t))) − sin(ωc t) sin(m sin(ωm t)))
x(t) is periodic in T = ω2π
, therefore cos(m sin(ωm t)) is periodic in T .
m
cos(m sin(ωm t))
1
t
0
−1
m=1
|ak |
0
10
20
30
23
40
50
60
k
Phase/Frequency Modulation
Find the Fourier transform of a PM/FM signal.
y(t) = cos(ωc t + mx(t)) = cos(ωc t + m sin(ωm t))
= cos(ωc t) cos(m sin(ωm t))) − sin(ωc t) sin(m sin(ωm t)))
x(t) is periodic in T = ω2π
, therefore cos(m sin(ωm t)) is periodic in T .
m
cos(m sin(ωm t))
1
t
0
−1
m=2
|ak |
0
10
20
30
24
40
50
60
k
Phase/Frequency Modulation
Find the Fourier transform of a PM/FM signal.
y(t) = cos(ωc t + mx(t)) = cos(ωc t + m sin(ωm t))
= cos(ωc t) cos(m sin(ωm t))) − sin(ωc t) sin(m sin(ωm t)))
x(t) is periodic in T = ω2π
, therefore cos(m sin(ωm t)) is periodic in T .
m
cos(m sin(ωm t))
1
t
0
−1
m=5
|ak |
0
10
20
30
25
40
50
60
k
Phase/Frequency Modulation
Find the Fourier transform of a PM/FM signal.
y(t) = cos(ωc t + mx(t)) = cos(ωc t + m sin(ωm t))
= cos(ωc t) cos(m sin(ωm t))) − sin(ωc t) sin(m sin(ωm t)))
x(t) is periodic in T = ω2π
, therefore cos(m sin(ωm t)) is periodic in T .
m
cos(m sin(ωm t))
1
t
0
−1
m = 10
|ak |
0
10
20
30
26
40
50
60
k
Phase/Frequency Modulation
Find the Fourier transform of a PM/FM signal.
y(t) = cos(ωc t + mx(t)) = cos(ωc t + m sin(ωm t))
= cos(ωc t) cos(m sin(ωm t))) − sin(ωc t) sin(m sin(ωm t)))
x(t) is periodic in T = ω2π
, therefore cos(m sin(ωm t)) is periodic in T .
m
cos(m sin(ωm t))
1
t
0
−1
m = 20
|ak |
0
10
20
30
27
40
50
60
k
Phase/Frequency Modulation
Find the Fourier transform of a PM/FM signal.
y(t) = cos(ωc t + mx(t)) = cos(ωc t + m sin(ωm t))
= cos(ωc t) cos(m sin(ωm t))) − sin(ωc t) sin(m sin(ωm t)))
x(t) is periodic in T = ω2π
, therefore cos(m sin(ωm t)) is periodic in T .
m
cos(m sin(ωm t))
1
t
0
−1
m = 30
|ak |
0
10
20
30
28
40
50
60
k
Phase/Frequency Modulation
Find the Fourier transform of a PM/FM signal.
y(t) = cos(ωc t + mx(t)) = cos(ωc t + m sin(ωm t))
= cos(ωc t) cos(m sin(ωm t))) − sin(ωc t) sin(m sin(ωm t)))
x(t) is periodic in T = ω2π
, therefore cos(m sin(ωm t)) is periodic in T .
m
cos(m sin(ωm t))
1
t
0
−1
m = 40
|ak |
0
10
20
30
29
40
50
60
k
Phase/Frequency Modulation
Find the Fourier transform of a PM/FM signal.
y(t) = cos(ωc t + mx(t)) = cos(ωc t + m sin(ωm t))
= cos(ωc t) cos(m sin(ωm t))) − sin(ωc t) sin(m sin(ωm t)))
x(t) is periodic in T = ω2π
, therefore cos(m sin(ωm t)) is periodic in T .
m
cos(m sin(ωm t))
1
t
0
−1
m = 50
|ak |
0
10
20
30
30
40
50
60
k
Phase/Frequency Modulation
Fourier transform of first part.
x(t) = sin(ωm t)
y(t) = cos(ωc t + mx(t)) = cos(ωc t + m sin(ωm t))
= cos(ωc t) cos(m sin(ωm t))) − sin(ωc t) sin(m sin(ωm t)))
'
v
"
ya (t)
|Ya (jω)|
m = 50
ω
ωc
ωc
100ωm
31
Phase/Frequency Modulation
Find the Fourier transform of a PM/FM signal.
y(t) = cos(ωc t + mx(t)) = cos(ωc t + m sin(ωm t))
= cos(ωc t) cos(m sin(ωm t))) − sin(ωc t) sin(m sin(ωm t)))
, therefore sin(m sin(ωm t)) is periodic in T .
x(t) is periodic in T = ω2π
m
m sin(ωm t)
m
t
0
−m
sin(m sin(ωm t))
1
increasing m
t
increasing m
0
−1
32
Phase/Frequency Modulation
Find the Fourier transform of a PM/FM signal.
y(t) = cos(ωc t + mx(t)) = cos(ωc t + m sin(ωm t))
= cos(ωc t) cos(m sin(ωm t))) − sin(ωc t) sin(m sin(ωm t)))
x(t) is periodic in T = ω2π
, therefore sin(m sin(ωm t)) is periodic in T .
m
sin(m sin(ωm t))
1
t
0
−1
m=0
|bk |
0
10
20
30
33
40
50
60
k
Phase/Frequency Modulation
Find the Fourier transform of a PM/FM signal.
y(t) = cos(ωc t + mx(t)) = cos(ωc t + m sin(ωm t))
= cos(ωc t) cos(m sin(ωm t))) − sin(ωc t) sin(m sin(ωm t)))
x(t) is periodic in T = ω2π
, therefore sin(m sin(ωm t)) is periodic in T .
m
sin(m sin(ωm t))
1
t
0
−1
m=1
|bk |
0
10
20
30
34
40
50
60
k
Phase/Frequency Modulation
Find the Fourier transform of a PM/FM signal.
y(t) = cos(ωc t + mx(t)) = cos(ωc t + m sin(ωm t))
= cos(ωc t) cos(m sin(ωm t))) − sin(ωc t) sin(m sin(ωm t)))
x(t) is periodic in T = ω2π
, therefore sin(m sin(ωm t)) is periodic in T .
m
sin(m sin(ωm t))
1
t
0
−1
m=2
|bk |
0
10
20
30
35
40
50
60
k
Phase/Frequency Modulation
Find the Fourier transform of a PM/FM signal.
y(t) = cos(ωc t + mx(t)) = cos(ωc t + m sin(ωm t))
= cos(ωc t) cos(m sin(ωm t))) − sin(ωc t) sin(m sin(ωm t)))
x(t) is periodic in T = ω2π
, therefore sin(m sin(ωm t)) is periodic in T .
m
sin(m sin(ωm t))
1
t
0
−1
m=5
|bk |
0
10
20
30
36
40
50
60
k
Phase/Frequency Modulation
Find the Fourier transform of a PM/FM signal.
y(t) = cos(ωc t + mx(t)) = cos(ωc t + m sin(ωm t))
= cos(ωc t) cos(m sin(ωm t))) − sin(ωc t) sin(m sin(ωm t)))
x(t) is periodic in T = ω2π
, therefore sin(m sin(ωm t)) is periodic in T .
m
sin(m sin(ωm t))
1
t
0
−1
m = 10
|bk |
0
10
20
30
37
40
50
60
k
Phase/Frequency Modulation
Find the Fourier transform of a PM/FM signal.
y(t) = cos(ωc t + mx(t)) = cos(ωc t + m sin(ωm t))
= cos(ωc t) cos(m sin(ωm t))) − sin(ωc t) sin(m sin(ωm t)))
x(t) is periodic in T = ω2π
, therefore sin(m sin(ωm t)) is periodic in T .
m
sin(m sin(ωm t))
1
t
0
−1
m = 20
|bk |
0
10
20
30
38
40
50
60
k
Phase/Frequency Modulation
Find the Fourier transform of a PM/FM signal.
y(t) = cos(ωc t + mx(t)) = cos(ωc t + m sin(ωm t))
= cos(ωc t) cos(m sin(ωm t))) − sin(ωc t) sin(m sin(ωm t)))
x(t) is periodic in T = ω2π
, therefore sin(m sin(ωm t)) is periodic in T .
m
sin(m sin(ωm t))
1
t
0
−1
m = 30
|bk |
0
10
20
30
39
40
50
60
k
Phase/Frequency Modulation
Find the Fourier transform of a PM/FM signal.
y(t) = cos(ωc t + mx(t)) = cos(ωc t + m sin(ωm t))
= cos(ωc t) cos(m sin(ωm t))) − sin(ωc t) sin(m sin(ωm t)))
x(t) is periodic in T = ω2π
, therefore sin(m sin(ωm t)) is periodic in T .
m
sin(m sin(ωm t))
1
t
0
−1
m = 40
|bk |
0
10
20
30
40
40
50
60
k
Phase/Frequency Modulation
Find the Fourier transform of a PM/FM signal.
y(t) = cos(ωc t + mx(t)) = cos(ωc t + m sin(ωm t))
= cos(ωc t) cos(m sin(ωm t))) − sin(ωc t) sin(m sin(ωm t)))
x(t) is periodic in T = ω2π
, therefore sin(m sin(ωm t)) is periodic in T .
m
sin(m sin(ωm t))
1
t
0
−1
m = 50
|bk |
0
10
20
30
41
40
50
60
k
Phase/Frequency Modulation
Fourier transform of second part.
x(t) = sin(ωm t)
y(t) = cos(ωc t + mx(t)) = cos(ωc t + m sin(ωm t))
= cos(ωc t) cos(m sin(ωm t))) − sin(ωc t) sin(m sin(ωm t)))
'
v
" '
v
"
ya (t)
yb (t)
|Yb (jω)|
m = 50
ω
ωc
ωc
100ωm
42
Phase/Frequency Modulation
Fourier transform.
x(t) = sin(ωm t)
y(t) = cos(ωc t + mx(t)) = cos(ωc t + m sin(ωm t))
= cos(ωc t) cos(m sin(ωm t))) − sin(ωc t) sin(m sin(ωm t)))
'
v
" '
v
"
ya (t)
yb (t)
|Y (jω)|
m = 50
ω
ωc
ωc
100ωm
43
Frequency Modulation
Wideband FM is useful because it is robust to noise.
AM: y1 (t) = (cos(ωm t) + 1.1) cos(ωc t)
t
FM: y3 (t) = cos(ωc t + m sin(ωm t))
t
FM generates a redundant signal that is resilient to additive noise.
44
Summary
Modulation is useful for matching signals to media.
Examples: commercial radio (AM and FM)
Close with unconventional application of modulation – in microscopy.
45
6.003 Microscopy
Dennis M. Freeman
Stanley S. Hong
Jekwan Ryu
Michael S. Mermelstein
Berthold K. P. Horn
Courtesy of Stanley Hong, Jekwan Ryu, Michael Mermelstein, and Berthold K. P. Horn. Used with permission.
46
6.003 Model of a Microscope
microscope
Microscope = low-pass filter
Courtesy of Stanley Hong, Jekwan Ryu, Michael Mermelstein, and Berthold K. P. Horn. Used with permission.
47
Phase-Modulated Microscopy
microscope
Courtesy of Stanley Hong, Jekwan Ryu, Michael Mermelstein, and Berthold K. P. Horn. Used with permission.
48
Courtesy of Stanley Hong, Jekwan Ryu, Michael Mermelstein, and Berthold K. P. Horn. Used with permission.
49
Courtesy of Stanley Hong, Jekwan Ryu, Michael Mermelstein, and Berthold K. P. Horn. Used with permission.
50
Courtesy of Stanley Hong, Jekwan Ryu, Michael Mermelstein, and Berthold K. P. Horn. Used with permission.
51
Courtesy of Stanley Hong, Jekwan Ryu, Michael Mermelstein, and Berthold K. P. Horn. Used with permission.
52
Courtesy of Stanley Hong, Jekwan Ryu, Michael Mermelstein, and Berthold K. P. Horn. Used with permission.
53
Courtesy of Stanley Hong, Jekwan Ryu, Michael Mermelstein, and Berthold K. P. Horn. Used with permission.
54
Courtesy of Stanley Hong, Jekwan Ryu, Michael Mermelstein, and Berthold K. P. Horn. Used with permission.
55
Courtesy of Stanley Hong, Jekwan Ryu, Michael Mermelstein, and Berthold K. P. Horn. Used with permission.
56
Courtesy of Stanley Hong, Jekwan Ryu, Michael Mermelstein, and Berthold K. P. Horn. Used with permission.
57
Courtesy of Stanley Hong, Jekwan Ryu, Michael Mermelstein, and Berthold K. P. Horn. Used with permission.
58
Courtesy of Stanley Hong, Jekwan Ryu, Michael Mermelstein, and Berthold K. P. Horn. Used with permission.
59
Courtesy of Stanley Hong, Jekwan Ryu, Michael Mermelstein, and Berthold K. P. Horn. Used with permission.
60
MIT OpenCourseWare
http://ocw.mit.edu
6.003 Signals and Systems
Fall 2011
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