Amplitude Modulation, Modulators, and Demodulators
ESE 488
Double Sideband Amplitude Modulation (AM)
vS(t)
VS(1+m)
VS
VS(1-m)
t
Figure 1 – Sinusoidal signal with a dc component
In double sideband modulation (the usual AM) a dc component is added to the signal
voltage before the signal is multiplied by a carrier. If the signal were a simple sinusoid, it would
have the form:
vS(t) = VS (1 + m cos st)
(1)
where VS is the dc component, s = 2fs is the signal frequency, and m is known as the
modulation index. This waveform is shown in Figure 1. To avoid distortion in recovering the
modulating signal with a simple demodulator, the modulation index, m, is constrained to lie in
the range zero to one.
In AM, the carrier signal has the form:
(2)
vC(t) = VC cos ct
where c is the carrier frequency in radians/sec. The carrier frequency, c, is usually much
greater than the signal frequency, s.
The modulated signal is then:
(3)
vm(t) = A vS(t) vC(t),
where A is a scale factor that depends on the equipment used for modulation. Using equations
(1) and (2) we can write
vm(t) = AVCVS (1 + m cos st) cos ct
v m ( t )  AVC VS cos c t 
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(4)
AVC VS m
[cos(c  s ) t  cos(c  s ) t ]
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(5)
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Equation (5) was obtained from equation (4) by using the trigonometric identity for the product
of two cosines.
Vm(t)
Vmax
AVCVS(1+m)
AVCVS
AVCVS(1-m)
Vmin
t
Figure 2 – Modulated signal vs. time
Figure 2 shows the modulated signal of equations (5) or (6) as a function of time. The
waveform above zero is produced by the positive values of cos ct while the values below zero
are the result of negative values of cos ct. Notice that the modulation index can be obtained by
measuring Vmax and Vmin. That is
m
Vmax  Vmin
Vmax  Vmin
(6)
Vm(f)
Carrier
AVCVS
Lower
Sideband
Upper
Sideband
mAVCVS/2
f
fc-fs
fc
fc+fs
Figure 3 – Frequency components of the modulated signal
Figure 3 shows the frequency components of the modulated double sideband signal. It is
evident from equation (5) that the modulated signal has a carrier component and upper and lower
sidebands at the sum and difference frequencies, (fs + fc) and (fs – fc). Note that the largest
values the sidebands can have relative to the carrier occurs when m = 1. This is referred to as
100% modulation and the sidebands are each half as large as the carrier.
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Double Sideband Suppressed Carrier Modulation
Vs(t)
+mVs
t
0
-mVs
Figure 4 – Sinusoidal signal with no dc component
A double sideband suppressed carrier signal has no dc component added to the signal so
that
(7)
vs(t) = mVS cos st
The modulated signal is the product of the modulating signal of equation (7) and the carrier with
the result
(8)
vm(t) = mAVCVS cos st cos ct
or
v m (t) 
mAVS VC
[cos( c  s ) t  cos( c  s ) t ]
2
(9)
Comparing this equation to equation (5) for the case of AM with carrier, we see that equation
(9) has no carrier frequency, c, term.
vm(t)
+mAVCVS
t
0
-mAVCVS
Figure 5 – Suppressed carrier modulated signal vs. time
Figure 5 shows the suppressed carrier signal of equation (9) as a function of time. The
positive lobe from 0 to 90o is produced by the product of a positive carrier and a positive signal,
while the positive lobe from 90o to 270o is produced by the product of a negative carrier and a
negative signal, etc. Notice that there is no envelope of the original modulating signal in the
modulated signal as there is in double sideband AM with carrier.
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The modulated signal as a function of frequency is shown in Figure 6. With no
component at the carrier frequency, the transmission power requirements are lower than for AM
with carrier.
Vm(f)
Lower
Sideband
Upper
Sideband
mAVCVS/2
f
fc-fs
fc+fs
Figure 6 – Frequency components of suppressed carrier signal
Demodulator
If the carrier signal is available at the demodulator, AM signals, either suppressed carrier
or with carrier, can be demodulated by multiplying the received AM signal by a local oscillator
sine wave (phase-locked to the carrier at the carrier frequency) and low pass filtering. For the
suppressed carrier case, for example, the product is
VI(t) = AVCVS m cos st cos2ct
(10)
or
AVC VS m
(11)
cos s t (1  cos 2c t )
2
The original signal is then recovered with appropriate low-pass filtering (i.e., removing the
product of the cos2st and cos2ct terms).
v I (t) 
Figure 7 shows the spectrum of the output of the multiplier in the receiver showing
baseband and double frequency components of (11).
Vm(f)
Lower
Sideband
Upper
Sideband
AVCVS/2
f
fs
2fc-fs
2fc+fs
Figure 7 – Frequency components after multiplication by carrier in receiver
See: http://en.wikipedia.org/wiki/Quadrature_amplitude_modulation for QAM.
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