13
Continuous-Time
Modulation
In this lecture, we begin the discussion of modulation. This is an important
concept in communication systems and, as we will see in Lecture 15, also provides the basis for converting between continuous-time and discrete-time signals. In its most general sense, modulation means using one signal to vary a
parameter of another signal. In communication systems, for example, if a
channel is particularly suited to transmission in a certain frequency range, the
information to be transmitted may be embedded in a carrier signal matched to
the channel. The mechanism by which the information is embedded is modulation; that is, the information to be transmitted is used to modulate some parameter of the carrier signal. In sinusoidal frequency modulation, for example,
the information is used to modulate the carrier frequency. In sinusoidal amplitude modulation, the carrier is sinusoidal at a frequency that the channel can
accommodate, and the information to be transmitted modulates the amplitude of this carrier. Furthermore, in communication systems, if many different signals are to be transmitted over the same channel, a technique referred
to asfrequency division multiplexing is often used. In this method each signal is used to modulate a carrier of a different frequency so that in the composite signal the information for each of the separate signals occupies nonoverlapping frequency bands.
The modulation property for Fourier transforms applies directly to amplitude modulation, that is, the interpretation in the frequency domain of the
result of multiplying a carrier signal by a modulating signal. From the modulation property we know that for amplitude modulation the spectrum of the
modulated output is the convolution of the spectra of the carrier and the modulating signal. When the carrier is either a complex exponential or a sinusoidal signal, the spectrum of the carrier is one or a pair of impulses and the
result of the convolution is then to shift the spectrum of the modulating signal
to a center frequency equal to the carrier frequency. Modulation with a single
complex exponential and with a sinusoidal signal are closely related.
With a complex exponential carrier, demodulation, i.e., recovery of the
original modulating signal, is relatively straightforward, basically involving
modulating a second time with the complex corjugate signal. With sinusoidal
13-1
Signals and Systems
13-2
amplitude modulation, demodulation consists of modulating again with a sinusoidal carrier followed by lowpass filtering to extract the original signal.
This form of demodulation is typically referred to as synchronous demodulation since it requires synchronization between the sinusoidal carrier signals in
the modulator and demodulator. However, by adding a constant to the modulating signal or equivalently injecting some carrier signal into the modulated
output, a simpler form of demodulator can be used. This is referred to as
asynchronousdemodulation and typically results in a less expensive demodulator. However, the fact that a carrier signal is injected into the modulated
signal represents an inefficiency of power transmission.
Suggested Reading
Section 7.0, Introduction, pages 447-448
Section 7.1, Continuous-Time Sinusoidal Amplitude Modulation, pages 449459
Section 7.2, Some Applications of Sinusoidal Amplitude Modulation, pages
459-464
Section 7.3, Single-Sideband Amplitude Modulation, pages 464-468
Continuous-Time Modulation
13-3
SINUSOIDAL AMPLITUDE
MODULATION
TRANSPARENCY
13.1
Sinusoidal amplitude
and frequency
modulation with a
sinusoidal carrier.
SINUSOIDAL FREQUENCY
A AA
MODULATION
t
TRANSPARENCY
13.2
Block diagram of
amplitude modulation
and some examples
of commonly used
carrier signals.
Signals and Systems
13-4
x(t) CMt 4xt
TRANSPARENCY
13.3
Spectra associated
with amplitude
modulation with a
complex exponential
carrier.
1ff (X(w)
*CMI~
YGw)
x(t)
-
C(w)
21e-ws
2e0*
t
+
t
Iz
OG 43.
I
I
W
0-twot + .)
liwe
X(w)
Y~w)
y(t)
+ 3M
(WC
- WM
4 1w. +"M)i
WM
~WM
00s (w~t +
6,)
Re jy(t)}
xKt)
X
y(t)
IMl" o
TRANSPARENCY
13.4
Representation of
amplitude modulation
with a complex
exponential carrier in
terms of amplitude
modulation with two
sinusoidal carriers
with a 90* phase
difference.
X()
WM
-- M
y"w)
y(t)
lY(C)+
Y* (-w)j
Re jy(t)
(Y()
jIm
jy(t)}
-
Y*(-w)
Continuous-Time Modulation
13-5
TRANSPARENCY
13.5
ejwct
e
Ideal lowpass
filter
The use of amplitude
modulation with a
complex exponential
carrier to implement
bandpass filtering with
a lowpass filter.
jct
H(w)
w(t)
X
y(t)
WO
X
f(t)
CO
X(o)
TRANSPARENCY
Y(o) W(o)
F-
I
A-
(JO
F(w)
(-we
-
WO
0)
(-Wd1
+ WO)
13.6
Spectra associated
with the system in
Transparency 13.5.
Signals and Systems
13-6
-------------
TRANSPARENCY
13.7
Equivalent frequency
response associated
with the system in
Transparency 13.5.
(-Wc -
(71
Wo0 )
(-wc
+
wO)
H, (W)
2
wo
H2(F
2
TRANSPARENCY
13.8
Representation of
amplitude modulation
with a complex
exponential carrier in
terms of amplitude
modulation with two
sinusoidal carriers
with a 90* phase
difference.
7
Continuous-Time Modulation
13-7
c(t)
cos (WCt + Oe)
=
2
ei(wt +0)
+
2
et + 60
e
X(C,)
TRANSPARENCY
13.9
Spectra associated
with amplitude
modulation with a
sinusoidal carrier.
- COM
C(o)
we
Ie
I
1
YC
weJC
)
C-je
l(Welo
Cos
(Wct
+
TRANSPARENCY
13.10
Demodulation of an
amplitude-modulated
signal with a
sinusoidal carrier.
0,)
I100
12 10
WC
(-
(-
-M)
+
WpM)W
C(W)
tie
tWeW
W(W)
i-2j9,I
I
1 2jOC
Signals and Systems
13-8
MODULATOR
x(t)
TRANSPARENCY
13.11
Block diagram of
sinusoidal amplitude
modulation and
demodulation.
y(t)
cos (w et +
0 C)
DEMODU LATOR
H (w)
2
x (t)
y (t)
w
w
-W
Lowpass filter
cos (oct + Oc)
WM < w <
(2oc
m
COS Wat
Ya (t)
TRANSPARENCY
13.12
Block diagram for
frequency division
multiplexing.
Xa (t)
COS ob t
Yb (t)
Xb (t)
Cos Wc t
Xc(W
w(t)
Continuous-Time Modulation
13-9
Xb (W)
X, (W)
-wM
-wM
(A
WM
X (
W
\M
M
M
TRANSPARENCY
13.13
Spectra illustrating
frequency division
multiplexing.
L
(
(A.)
-Wb
W(W)
Ar7inK> K>r7~~A
-
C
-Cb
-
Wa
Wa
Demultiplexing Bandpass
filter
Lb
Mc
L
TRANSPARENCY
13.14
Demultiplexing and
demodulation of a
frequency division
multiplexed signal.
Demodulation
COS Wat
Lowpass
filter
Xa (t)
Signals and Systems
13-10
Cos
TRANSPARENCY
13.15
Demodulation of an
amplitude-modulated
signal with a
sinusoidal carrier.
[Transparency 13.10
repeated]
(WC
t + 0c)
Y (W)
1 el6
j2i
- 11
(W,-
M) W. (W. + WM)
W
C(I
ciw)
Weit
re~lI
W(W)
I~O
-2jO
-
cos
(ct
+
w,
I
W
L
M2
0c)
TRANSPARENCY
13.16
Effect of loss of
synchronization
in phase in a
synchronous
sinusoidal amplitude
modulation/demodulation system.
(oe -
Lowpass filter
cos (oct + 4c)
w(t) = x(t) Cos (Wmt + 6,) cos (ot +
=
[cos (0c ~ oc)]x(t) +-I- x(t) cos (2wct +
lowpass component
Oc+
#c)
+c)]x(t)
Continuous-Time Modulation
13-11
TRANSPARENCY
13.17
A simple system and
associated waveform
for an asynchronous
demodulation system.
y(t) - IA + x(tI cos wCt
Cos ('t
AAAW4-A-yyV.
vA
AT
ANA~
IY--V.N,Lv
~NA
A
TRANSPARENCY
13.18
Modulator associated
with asynchronous
sinusoidal amplitude
modulation. For such
a system the carrier
must be irjected into
the output. This
transparency shows
time waveforms for
the asynchronous
modulation system.
Signals and Systems
13-12
y(t) -
A+ x (t)I cos
oct
TRANSPARENCY
13.19
Frequency spectra
associated with an
asynchronous
modulation system.
cos (",t
irA
iTA
1
2
X(W)
TRANSPARENCY
13.20
WM
~W M
Single-sideband
modulation in which
only the upper
sidebands are
retained.
Y(W)
sideband sideband
sideband sideband
Y, (w)
-we
-C
W
Continuous-Time Modulation
13-13
TRANSPARENCY
13.21
Single-sideband
modulation in which
only the lower
sidebands are
retained.
v (t)
TRANSPARENCY
13.22
The use of a highpass
filter to obtain a
single-sideband signal.
11
2
c
W
H(w)
-Loc
cW
1- -
A
2
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Resource: Signals and Systems
Professor Alan V. Oppenheim
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