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+st)cos2π
f
c
t
1
s
t
2
f
c
t

(1)
Thus,

st=(1+st)cos2π
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∫−∞∞Sfei2π(f+
f
c
)tdf+12∫−∞∞Sfei2π(f−
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.