Aula 3

Prévia do material em texto

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.
 
x
1
(t)
x
2
(t)
x
3
(t)
z
1
(t)
z
2
(t) z(t)
y(t)
z
3
(t)
cos w
1
t
cos w
2
t
cos w
c
t
cos w
3
t
LPF
3
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(ωct) 
Phase Modulation (PM): y2(t) = cos(ωct + kx(t)) tFrequency Modulation (FM): y3(t) = cos ωct + k −∞ x(τ )dτ
PM: signal modulates instantaneous phase of the carrier. 
y2(t) = cos(ωct + kx(t)) 
FM: signal modulates instantaneous frequency of carrier. t 
y3(t) = cos ωct + k x(τ)dτ ' −∞v " 
φ(t)
d 
ωi(t) = ωc + φ(t) = ωc + kx(t)dt 
5
 
 
Frequency Modulation
 
sin(ωmt)) 
Advantages of FM: 
• constant power 
Compare AM to FM for x(t) = cos(ωmt). 
AM: y1(t) = x(t) + C cos(ωct) = (cos(ωmt) + 1.1) cos(ωct) 
t
FM: y3(t) = cos ωct + k −∞ x(τ )dτ = cos(ωct +
t k 
ωm 
t
• 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. 
t 
y3(t) = cos ωct + 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 t 
y3(t) = cos ωct + k x(τ)dτ
−∞0 0 t t 
= cos(ωct) × cos k x(τ)dτ − sin(ωct) × sin k x(τ )dτ
−∞ −∞ 
If k → 0 then0 t 
cos k x(τ)dτ → 1 
−∞0 t t 
sin k x(τ )dτ → k x(τ)dτ 
−∞ −∞0 t 
y3(t) ≈ cos(ωct) − sin(ωct) × 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
∫
∫ ∫
∫
∫ ∫
∫
1
0
−1
1 sin(ωmt)
t
1
0
−1
cos(1 sin(ωmt))
t
Phase/Frequency Modulation
 
Find the Fourier transform of a PM/FM signal. 
y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) 
= cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 
x(t) is periodic in T = 2π , therefore cos(m sin(ωmt)) is periodic in T .ωm 
9
2
0
−2
2 sin(ωmt)
t
1
0
−1
cos(2 sin(ωmt))
t
Phase/Frequency Modulation
 
Find the Fourier transform of a PM/FM signal. 
y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) 
= cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 
x(t) is periodic in T = 2π , therefore cos(m sin(ωmt)) is periodic in T .ωm 
10
3
0
−3
3 sin(ωmt)
t
1
0
−1
cos(3 sin(ωmt))
t
Phase/Frequency Modulation
 
Find the Fourier transform of a PM/FM signal. 
y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) 
= cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 
x(t) is periodic in T = 2π , therefore cos(m sin(ωmt)) is periodic in T .ωm 
11
4
0
−4
4 sin(ωmt)
t
1
0
−1
cos(4 sin(ωmt))
t
Phase/Frequency Modulation
 
Find the Fourier transform of a PM/FM signal. 
y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) 
= cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 
x(t) is periodic in T = 2π , therefore cos(m sin(ωmt)) is periodic in T .ωm 
12
5
0
−5
5 sin(ωmt)
t
1
0
−1
cos(5 sin(ωmt))
t
Phase/Frequency Modulation
 
Find the Fourier transform of a PM/FM signal. 
y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) 
= cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 
x(t) is periodic in T = 2π , therefore cos(m sin(ωmt)) is periodic in T .ωm 
13
6
0
−6
6 sin(ωmt)
t
1
0
−1
cos(6 sin(ωmt))
t
Phase/Frequency Modulation
 
Find the Fourier transform of a PM/FM signal. 
y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) 
= cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 
x(t) is periodic in T = 2π , therefore cos(m sin(ωmt)) is periodic in T .ωm 
14
7
0
−7
7 sin(ωmt)
t
1
0
−1
cos(7 sin(ωmt))
t
Phase/Frequency Modulation
 
Find the Fourier transform of a PM/FM signal. 
y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) 
= cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 
x(t) is periodic in T = 2π , therefore cos(m sin(ωmt)) is periodic in T .ωm 
15
8
0
−8
8 sin(ωmt)
t
1
0
−1
cos(8 sin(ωmt))
t
Phase/Frequency Modulation
 
Find the Fourier transform of a PM/FM signal. 
y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) 
= cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 
x(t) is periodic in T = 2π , therefore cos(m sin(ωmt)) is periodic in T .ωm 
16
9
0
−9
9 sin(ωmt)
t
1
0
−1
cos(9 sin(ωmt))
t
Phase/Frequency Modulation
 
Find the Fourier transform of a PM/FM signal. 
y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) 
= cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 
x(t) is periodic in T = 2π , therefore cos(m sin(ωmt)) is periodic in T .ωm 
17
10
0
−10
10 sin(ωmt)
t
1
0
−1
cos(10 sin(ωmt))
t
Phase/Frequency Modulation
 
Find the Fourier transform of a PM/FM signal. 
y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) 
= cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 
x(t) is periodic in T = 2π , therefore cos(m sin(ωmt)) is periodic in T .ωm 
18
20
0
−20
20 sin(ωmt)
t
1
0
−1
cos(20 sin(ωmt))
t
Phase/Frequency Modulation
 
Find the Fourier transform of a PM/FM signal. 
y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) 
= cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 
x(t) is periodic in T = 2π , therefore cos(m sin(ωmt)) is periodic in T .ωm 
19
50
0
−50
50 sin(ωmt)
t
1
0
−1
cos(50 sin(ωmt))
t
Phase/Frequency Modulation
 
Find the Fourier transform of a PM/FM signal. 
y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) 
= cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 
x(t) is periodic in T = 2π , therefore cos(m sin(ωmt)) is periodic in T .ωm 
20
m
0
−m
m sin(ωmt)
t
1
0
−1
cos(m sin(ωmt))
t
increasing m
Phase/Frequency Modulation
 
Find the Fourier transform of a PM/FM signal. 
y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) 
= cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 
x(t) is periodic in T = 2π , therefore cos(m sin(ωmt)) is periodic in T .ωm 
21
Phase/Frequency Modulation
 
Find the Fourier transform of a PM/FM signal. 
y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) 
= cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 
x(t) is periodic in T = 2π ωm , therefore cos(m sin(ωmt)) is periodic in T . 
1
0
−1
cos(m sin(ωmt))
t
|ak|
k
0 10 20 30 40 50 60
m = 0
22
Phase/FrequencyModulation
 
Find the Fourier transform of a PM/FM signal. 
y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) 
= cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 
x(t) is periodic in T = 2π ωm , therefore cos(m sin(ωmt)) is periodic in T . 
1
0
−1
cos(m sin(ωmt))
t
|ak|
k
0 10 20 30 40 50 60
m = 1
23
Phase/Frequency Modulation
 
Find the Fourier transform of a PM/FM signal. 
y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) 
= cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 
x(t) is periodic in T = 2π ωm , therefore cos(m sin(ωmt)) is periodic in T . 
1
0
−1
cos(m sin(ωmt))
t
|ak|
k
0 10 20 30 40 50 60
m = 2
24
Phase/Frequency Modulation
 
Find the Fourier transform of a PM/FM signal. 
y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) 
= cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 
x(t) is periodic in T = 2π ωm , therefore cos(m sin(ωmt)) is periodic in T . 
1
0
−1
cos(m sin(ωmt))
t
|ak|
k
0 10 20 30 40 50 60
m = 5
25
Phase/Frequency Modulation
 
Find the Fourier transform of a PM/FM signal. 
y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) 
= cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 
x(t) is periodic in T = 2π ωm , therefore cos(m sin(ωmt)) is periodic in T . 
1
0
−1
cos(m sin(ωmt))
t
|ak|
k
0 10 20 30 40 50 60
m = 10
26
Phase/Frequency Modulation
 
Find the Fourier transform of a PM/FM signal. 
y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) 
= cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 
x(t) is periodic in T = 2π ωm , therefore cos(m sin(ωmt)) is periodic in T . 
1
0
−1
cos(m sin(ωmt))
t
|ak|
k
0 10 20 30 40 50 60
m = 20
27
Phase/Frequency Modulation
 
Find the Fourier transform of a PM/FM signal. 
y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) 
= cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 
x(t) is periodic in T = 2π ωm , therefore cos(m sin(ωmt)) is periodic in T . 
1
0
−1
cos(m sin(ωmt))
t
|ak|
k
0 10 20 30 40 50 60
m = 30
28
Phase/Frequency Modulation
 
Find the Fourier transform of a PM/FM signal. 
y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) 
= cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 
x(t) is periodic in T = 2π ωm , therefore cos(m sin(ωmt)) is periodic in T . 
1
0
−1
cos(m sin(ωmt))
t
|ak|
k
0 10 20 30 40 50 60
m = 40
29
Phase/Frequency Modulation
 
Find the Fourier transform of a PM/FM signal. 
y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) 
= cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 
x(t) is periodic in T = 2π ωm , therefore cos(m sin(ωmt)) is periodic in T . 
1
0
−1
cos(m sin(ωmt))
t
|ak|
k
0 10 20 30 40 50 60
m = 50
30
|Ya(jω)|
ω
ωcωc
100ωm
m = 50
Phase/Frequency Modulation
 
Fourier transform of first part. 
x(t) = sin(ωmt) 
y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) 
= cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt)))' v " 
ya(t) 
31
m
0
−m
m sin(ωmt)
t
1
0
−1
sin(m sin(ωmt))
t
increasing m
increasing m
Phase/Frequency Modulation
 
Find the Fourier transform of a PM/FM signal. 
y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) 
= cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 
x(t) is periodic in T = 2π , therefore sin(m sin(ωmt)) is periodic in T .ωm 
32
Phase/Frequency Modulation
 
Find the Fourier transform of a PM/FM signal. 
y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) 
= cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 
x(t) is periodic in T = 2π ωm , therefore sin(m sin(ωmt)) is periodic in T . 
1
0
−1
sin(m sin(ωmt))
t
|bk|
k
0 10 20 30 40 50 60
m = 0
33
Phase/Frequency Modulation
 
Find the Fourier transform of a PM/FM signal. 
y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) 
= cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 
x(t) is periodic in T = 2π ωm , therefore sin(m sin(ωmt)) is periodic in T . 
1
0
−1
sin(m sin(ωmt))
t
|bk|
k
0 10 20 30 40 50 60
m = 1
34
Phase/Frequency Modulation
 
Find the Fourier transform of a PM/FM signal. 
y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) 
= cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 
x(t) is periodic in T = 2π ωm , therefore sin(m sin(ωmt)) is periodic in T . 
1
0
−1
sin(m sin(ωmt))
t
|bk|
k
0 10 20 30 40 50 60
m = 2
35
Phase/Frequency Modulation
 
Find the Fourier transform of a PM/FM signal. 
y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) 
= cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 
x(t) is periodic in T = 2π ωm , therefore sin(m sin(ωmt)) is periodic in T . 
1
0
−1
sin(m sin(ωmt))
t
|bk|
k
0 10 20 30 40 50 60
m = 5
36
Phase/Frequency Modulation
 
Find the Fourier transform of a PM/FM signal. 
y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) 
= cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 
x(t) is periodic in T = 2π ωm , therefore sin(m sin(ωmt)) is periodic in T . 
1
0
−1
sin(m sin(ωmt))
t
|bk|
k
0 10 20 30 40 50 60
m = 10
37
Phase/Frequency Modulation
 
Find the Fourier transform of a PM/FM signal. 
y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) 
= cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 
x(t) is periodic in T = 2π ωm , therefore sin(m sin(ωmt)) is periodic in T . 
1
0
−1
sin(m sin(ωmt))
t
|bk|
k
0 10 20 30 40 50 60
m = 20
38
Phase/Frequency Modulation
 
Find the Fourier transform of a PM/FM signal. 
y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) 
= cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 
x(t) is periodic in T = 2π ωm , therefore sin(m sin(ωmt)) is periodic in T . 
1
0
−1
sin(m sin(ωmt))
t
|bk|
k
0 10 20 30 40 50 60
m = 30
39
Phase/Frequency Modulation
 
Find the Fourier transform of a PM/FM signal. 
y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) 
= cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 
x(t) is periodic in T = 2π ωm , therefore sin(m sin(ωmt)) is periodic in T . 
1
0
−1
sin(m sin(ωmt))
t
|bk|
k
0 10 20 30 40 50 60
m = 40
40
Phase/Frequency Modulation
 
Find the Fourier transform of a PM/FM signal. 
y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) 
= cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 
x(t) is periodic in T = 2π ωm , therefore sin(m sin(ωmt)) is periodic in T . 
1
0
−1
sin(m sin(ωmt))
t
|bk|
k
0 10 20 30 40 50 60
m = 50
41
' v "
ya(t)
|Yb(jω)|
ω
ωcωc
100ωm
m = 50
Phase/Frequency Modulation
 
Fourier transform of second part. 
x(t) = sin(ωmt) 
y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) 
= cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt)))' v " 
yb(t) 
42
|Y (jω)|
ω
ωcωc
100ωm
m = 50
Phase/Frequency Modulation
 
Fourier transform. 
x(t) = sin(ωmt) 
y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) 
= cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt)))' v " ' v " 
ya(t) yb(t) 
43
Frequency Modulation
 
Wideband FM is useful because it is robust to noise. 
AM: y1(t) = (cos(ωmt) + 1.1) cos(ωct) 
t
FM: y3(t) = cos(ωct + m sin(ωmt)) 
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 
46
Courtesy of Stanley Hong, Jekwan Ryu,Michael Mermelstein, and Berthold K. P. Horn. Used with permission.
6.003 Model of a Microscope 
microscope 
Microscope = low-pass filter 
47
Courtesy of Stanley Hong, Jekwan Ryu, Michael Mermelstein, and Berthold K. P. Horn. Used with permission.
Phase-Modulated Microscopy 
microscope 
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.
many frequencies + many orientations = many images
low resolution high resolution
wx
wy
wx
wy
wx
wy
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
Courtesy of Stanley Hong, Jekwan Ryu, Michael Mermelstein, and Berthold K. P. Horn. Used with permission.
MIT OpenCourseWare
http://ocw.mit.edu
6.003 Signals and Systems
Fall 2011
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.