In communications, a very important operation on a signal
st
s
t
is to amplitude modulate it. Using this operation
more as an example rather than elaborating the communications
aspects here, we want to compute the Fourier
transform—the spectrum—of
1+stcos2π
f
c
t
1
s
t
2
f
c
t
(1)
Thus,
st=1+stcos2π
f
c
t=cos2π
f
c
t+stcos2π
f
c
t
s
t
1
s
t
2
f
c
t
2
f
c
t
s
t
2
f
c
t
(2)
For the spectrum of
cos2π
f
c
t
2
f
c
t
, we use the Fourier series. Its period is
1
f
c
1
f
c
, and its only nonzero Fourier coefficients are
c
±
1
=12
c
±
1
1
2
.
The second term is
not periodic unless
st
s
t
has the same period as the sinusoid. Using Euler's relation,
the spectrum of the second term can be derived as
stcos2π
f
c
t=∫-∞∞Sfⅇⅈ2πftdfcos2π
f
c
t=12∫-∞∞Sfⅇⅈ2πf+
f
c
tdf+12∫-∞∞Sfⅇⅈ2πf-
f
c
tdf=12∫-∞∞Sf-
f
c
ⅇⅈ2πftdf+12∫-∞∞Sf+
f
c
ⅇⅈ2πftdf=∫-∞∞Sf-
f
c
+Sf+
f
c
2ⅇⅈ2πftdf
s
t
2
f
c
t
f
S
f
2
f
t
2
f
c
t
1
2
f
S
f
2
f
f
c
t
1
2
f
S
f
2
f
f
c
t
1
2
f
S
f
f
c
2
f
t
1
2
f
S
f
f
c
2
f
t
f
S
f
f
c
S
f
f
c
2
2
f
t
(3)
Exploiting the uniqueness property of the Fourier transform, we have
Fstcos2π
f
c
t=Sf-
f
c
+Sf+
f
c
2
F
s
t
2
f
c
t
S
f
f
c
S
f
f
c
2
(4)
This component of the spectrum consists of the original
signal's spectrum delayed and advanced
in
frequency. The spectrum of the amplitude modulated
signal is shown in
Figure 1.
Note how in this figure the signal
st
s
t
is defined in the frequency domain. To find its time domain
representation, we simply use the inverse Fourier transform.
What is the signal
st
s
t
that corresponds to the spectrum shown in the upper panel of
Figure 1?
The signal is the inverse Fourier transform
of the triangularly shaped spectrum, and equals
st=WsinπWtπWt2
s
t
W
W
t
W
t
2
What is the power in
xt
x
t
, the amplitude-modulated signal? Try the calculation in
both the time and frequency domains.
The result is most easily found in the spectrum's formula:
the power in the signal-related part of
xt
x
t
is half the power of the signal
st
s
t
.
In this example, we call the signal
st
s
t
a baseband signal because its power is contained at
low frequencies. Signals such as speech and the Dow Jones
averages are baseband signals. The baseband signal's
bandwidth equals
W
W
, the highest frequency at which it has power. Since
xt
x
t
's spectrum is confined to a frequency band not close to the
origin (we assume
f
c
≫W
≫
f
c
W
), we have a bandpass signal. The bandwidth of a
bandpass signal is not its highest
frequency, but the range of positive frequencies where the
signal has power. Thus, in this example, the bandwidth is
2W
Hz
2
W
Hz. Why a signal's bandwidth should
depend on its spectral shape will become clear once we develop
communications systems.
