Post on 21-Oct-2020
6.003: Signals and Systems
Modulation
May 6, 2010
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
Amplitude Modulation
Amplitude modulation can be used to match audio frequencies to
radio frequencies. It allows parallel transmission of multiple channels.
x1(t)
z1(t)
cos 1t
z2(t) z(t)
z3(t)
cos 2t cos ct
cos 3t
LPFx2(t) y(t)
x3(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 invented wide-band FM.
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Amplitude, Phase, and Frequency Modulation
There are many ways to embed a “message” in a carrier. Here are
three.
Amplitude Modulation (AM): y1(t) = x(t) cos(ωct)
Phase Modulation (PM): y2(t) = cos(ωct + kx(t)) � tFrequency Modulation (FM): y3(t) = cos ωct + k −∞ x(τ)dτ
� �� �
Frequency Modulation
In FM, the signal modulates the instantaneous carrier frequency.
� � t � y3(t) = cos ωct + k x(τ)dτ
−∞ φ(t)
ωi(t) = ωc + dφ(t) = ωc + kx(t)dt
Frequency Modulation
Compare AM to FM for x(t) = cos(ωmt).
AM: y1(t) = (cos(ωmt) + 1.1) cos(ωct)
t
FM: y3(t) = cos(ωct + m sin(ωmt))
t
Advantages of FM:
• constant power • no need to transmit carrier (unless DC important)
• bandwidth?
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� � � � � � �
� � �
Frequency Modulation
Early investigators thought that narrowband FM could have arbitrar
ily narrow bandwidth, allowing more channels than AM. Wrong! t
y3(t) = cos ωct + k x(τ )dτ −∞ � � � � � �t t
= cos(ωct) × cos k x(τ )dτ − sin(ωct) × sin k x(τ )dτ −∞ −∞
If k → 0 then t
cos k x(τ)dτ → 1 −∞ t t
sin k x(τ)dτ → k x(τ )dτ −∞ −∞
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 bandwidth)
Phase/Frequency Modulation
Find the Fourier transform of a PM signal.
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)))
2πx(t) is periodic in T = ωm , therefore cos(m sin(ωmt)) is periodic in T .
m sin(ωmt) m
0
−m
t
cos(m sin(ωmt))
t increasing m
1
0
−1
Phase/Frequency Modulation
Find the Fourier transform of a PM signal.
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)))
2πx(t) is periodic in T = ωm , therefore cos(m sin(ωmt)) is periodic in T .
cos(m sin(ωmt))1
0 t
−1 m = 0
|ak|
k0 10 20 30 40 50 60
Phase/Frequency Modulation
Find the Fourier transform of a PM signal.
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)))
2πx(t) is periodic in T = ωm , therefore cos(m sin(ωmt)) is periodic in T .
cos(m sin(ωmt))1
0 t
−1 m = 1
|ak|
k0 10 20 30 40 50 60
Phase/Frequency Modulation
Find the Fourier transform of a PM signal.
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)))
2πx(t) is periodic in T = ωm , therefore cos(m sin(ωmt)) is periodic in T .
cos(m sin(ωmt))1
0 t
−1 m = 2
|ak|
k0 10 20 30 40 50 60
Phase/Frequency Modulation
Find the Fourier transform of a PM signal.
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)))
2πx(t) is periodic in T = ωm , therefore cos(m sin(ωmt)) is periodic in T .
cos(m sin(ωmt))1
0 t
−1 m = 5
|ak|
k0 10 20 30 40 50 60
Phase/Frequency Modulation
Find the Fourier transform of a PM signal.
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)))
2πx(t) is periodic in T = ωm , therefore cos(m sin(ωmt)) is periodic in T .
cos(m sin(ωmt))1
0 t
−1 m = 10
|ak|
k0 10 20 30 40 50 60
Phase/Frequency Modulation
Find the Fourier transform of a PM signal.
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)))
2πx(t) is periodic in T = ωm , therefore cos(m sin(ωmt)) is periodic in T .
cos(m sin(ωmt))1
0 t
−1 m = 20
|ak|
k0 10 20 30 40 50 60
Phase/Frequency Modulation
Find the Fourier transform of a PM signal.
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)))
2πx(t) is periodic in T = ωm , therefore cos(m sin(ωmt)) is periodic in T .
cos(m sin(ωmt))1
0 t
−1 m = 30
|ak|
k0 10 20 30 40 50 60
Phase/Frequency Modulation
Find the Fourier transform of a PM signal.
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)))
2πx(t) is periodic in T = ωm , therefore cos(m sin(ωmt)) is periodic in T .
cos(m sin(ωmt))1
0 t
−1 m = 40
|ak|
k0 10 20 30 40 50 60
Phase/Frequency Modulation
Find the Fourier transform of a PM signal.
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)))
2πx(t) is periodic in T = ωm , therefore cos(m sin(ωmt)) is periodic in T .
cos(m sin(ωmt))1
0 t
−1 m = 50
|ak|
k0 10 20 30 40 50 60
� �� �
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)))
ya(t)
|Ya(jω)| m = 50
ω ωc ωc
50ωm
increasing m
Phase/Frequency Modulation
Find the Fourier transform of a PM signal.
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)))
2πx(t) is periodic in T = ωm , therefore sin(m sin(ωmt)) is periodic in T .
m sin(ωmt) m
0
−m
t
sin(m sin(ωmt))1
0
−1
t
increasing m
Phase/Frequency Modulation
Find the Fourier transform of a PM signal.
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)))
2πx(t) is periodic in T = ωm , therefore sin(m sin(ωmt)) is periodic in T .
sin(m sin(ωmt))1
0 t
−1 m = 0
|bk|
k0 10 20 30 40 50 60
Phase/Frequency Modulation
Find the Fourier transform of a PM signal.
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)))
2πx(t) is periodic in T = ωm , therefore sin(m sin(ωmt)) is periodic in T .
sin(m sin(ωmt))1
0 t
−1 m = 1
|bk|
k0 10 20 30 40 50 60
Phase/Frequency Modulation
Find the Fourier transform of a PM signal.
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)))
2πx(t) is periodic in T = ωm , therefore sin(m sin(ωmt)) is periodic in T .
sin(m sin(ωmt))1
0 t
−1 m = 2
|bk|
k0 10 20 30 40 50 60
Phase/Frequency Modulation
Find the Fourier transform of a PM signal.
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)))
2πx(t) is periodic in T = ωm , therefore sin(m sin(ωmt)) is periodic in T .
sin(m sin(ωmt))1
0 t
−1 m = 5
|bk|
k0 10 20 30 40 50 60
Phase/Frequency Modulation
Find the Fourier transform of a PM signal.
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)))
2πx(t) is periodic in T = ωm , therefore sin(m sin(ωmt)) is periodic in T .
sin(m sin(ωmt))1
0 t
−1 m = 10
|bk|
k0 10 20 30 40 50 60
Phase/Frequency Modulation
Find the Fourier transform of a PM signal.
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)))
2πx(t) is periodic in T = ωm , therefore sin(m sin(ωmt)) is periodic in T .
0 t
sin(m sin(ωmt))1
−1 m = 20
|bk|
k0 10 20 30 40 50 60
Phase/Frequency Modulation
Find the Fourier transform of a PM signal.
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)))
2πx(t) is periodic in T = ωm , therefore sin(m sin(ωmt)) is periodic in T .
sin(m sin(ωmt))1
0 t
−1 m = 30
|bk|
k0 10 20 30 40 50 60
Phase/Frequency Modulation
Find the Fourier transform of a PM signal.
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)))
2πx(t) is periodic in T = ωm , therefore sin(m sin(ωmt)) is periodic in T .
sin(m sin(ωmt))1
0 t
−1 m = 40
|bk|
k0 10 20 30 40 50 60
Phase/Frequency Modulation
Find the Fourier transform of a PM signal.
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)))
2πx(t) is periodic in T = ωm , therefore sin(m sin(ωmt)) is periodic in T .
sin(m sin(ωmt))1
0 t
−1 m = 50
|bk|
k0 10 20 30 40 50 60
� �� �
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))) yb(t)
|Yb(jω)| m = 50
ω
ωc ωc
50ωm
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))) � �� � � �� � ya(t) yb(t)
|Y (jω)| m = 50
ω
ωc ωc
50ωm
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 very redundant signal, which is resilient to additive
noise.
Summary
Modulation is useful for matching signals to media.
Examples: commercial radio (AM and FM)
Close with unconventional application of modulation – in microscopy.
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6.003 Signals and Systems Spring 2010
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