Especially for wireless channels, like commercial radio and
television, but also for wireline systems like cable television,
an analog message signal must be modulated: The
transmitted signal's spectrum occurs at much higher frequencies
than those occupied by the signal.
We use analog communication techniques for analog message
signals, like music, speech, and television. Transmission and
reception of analog signals using analog results in an
inherently noisy received signal (assuming the channel adds
noise, which it almost certainly does).
The key idea of modulation is to affect the amplitude, frequency
or phase of what is known as the
carrier
sinusoid. Frequency modulation (FM) and less frequently used
phase modulation (PM) are not discussed here; we focus on
amplitude modulation (AM). The amplitude modulated message
signal has the form
xt=
A
c
(1+mt)cos2π
f
c
t
x
t
A
c
1
m
t
2
f
c
t
(1) where
f
c
f
c
is the
carrier frequency and
A
c
A
c
the
carrier amplitude. Also, the signal's
amplitude is assumed to be less than one:
|mt|<1
m
t
1
. From our previous exposure to amplitude modulation
(see the
Fourier Transform example), we know that the
transmitted signal's spectrum occupies the frequency range
fc−W
fc+W
fc
W
fc
W
, assuming the signal's bandwidth is
WW Hz (see the
figure). The carrier
frequency is usually much larger than the signal's highest
frequency:
fc≫W
≫
fc
W
, which means that the transmitter antenna and carrier
frequency are chosen jointly during the design process.
Ignoring the attenuation and noise introduced by the channel for
the moment, reception of an amplitude modulated signal is quite
easy (see (Reference)).
The so-called coherent receiver multiplies the
input signal by a sinusoid and lowpass-filters the result (Figure 1).
m
^
t=LPFxtcos2π
f
c
t=LPF
A
c
(1+mt)cos22π
f
c
t
m
^
t
LPF
x
t
2
f
c
t
LPF
A
c
1
m
t
2
f
c
t
2
(2)
Because of our trigonometric identities, we know that
cos22πfct=12(1+cos2π2fct)
2
fc
t
2
1
2
1
2
2
fc
t
(3)
At this point, the message signal is multiplied by a constant
and a sinusoid at twice the carrier frequency. Multiplication by
the constant term returns the message signal to baseband (where
we want it to be!) while multiplication by the double-frequency
term yields a very high frequency signal. The lowpass filter
removes this high-frequency signal, leaving only the baseband
signal. Thus, the received signal is
m^
t=Ac2(1+mt)
m^
t
Ac
2
1
m
t
(4)
This derivation relies solely on the time domain; derive the
same result in the frequency domain. You won't need the
trigonometric identity with this approach.
The signal-related portion of the transmitted spectrum is
given by
Xf=12Mf−
f
c
+12Mf+
f
c
X
f
1
2
M
f
f
c
1
2
M
f
f
c
.
Multiplying at the receiver by the carrier shifts this
spectrum to
fc
fc
and to
−fc
fc
, and scales the result by half.
12Xf−
f
c
+12Xf+
f
c
=14(Mf−2
f
c
+Mf)+14(Mf+2
f
c
+Mf)=14Mf−2
f
c
+12Mf+14Mf+2
f
c
1
2
X
f
f
c
1
2
X
f
f
c
1
4
M
f
2
f
c
M
f
1
4
M
f
2
f
c
M
f
1
4
M
f
2
f
c
1
2
M
f
1
4
M
f
2
f
c
(5)
The signal components centered at twice the carrier frequency
are removed by the lowpass filter, while the baseband signal
Mf
M
f
emerges.
Because it is so easy to remove the constant term by electrical
means—we insert a capacitor in series with the receiver's
output—we typically ignore it and concentrate on the signal
portion of the receiver's output when calculating
signal-to-noise ratio.
"Electrical Engineering Digital Processing Systems in Braille."