Amplitude Modulation
Cuthbert Nyack
This was the first type of modulation used for communicating signals
from one point to another and is still the simplest to understand.
The signal can be written as:-
v = ac (1 +
m cos wmt)
cos wct
This represents a signal at frequency
wc
whose amplitude is modulated by another frequency
wm.
m = am/ac is the modulation index.
To find the frequency spectrum of the am signal the
above expression can be rewritten as a sum of signals
of constant amplitude:-
v(t) = ac{cos wct
+ m/2(cos(wc
+ wm)t
+ cos(wc
- wm)t)}
Above expression shows that the frequency spectrum consists of
3 components at frequencies
wc,
wc
+ wm and
wc
- wm.
The percent of the power transmitted which is in the carrier
is given by
Pc = 100/(1 + m * m/2) and varies from 100% for m = 0
to 66.66% for m = 1. This is considered to be one of the
disadvantages of AM since the carrier is a sine wave and
contains no information.
If the modulating index becomes greater than 1 then the expression
for the AM signal amplitude (1 + m cos wmt)
can go negative. In practical AM systems The upper amplitude of the envelope will be limited to
zero and the modulating index to one. However this needs to be
modified for modulation with many sines.
The applet below for Fn = 0 shows how an AM signal is related to the carrier and the modulating signals.
An AM signal can be described by any combination of sines and cosines. As mentioned above:-
v = ac (1 +
m cos wmt)
cos wct = ac{cos wct
+ m/2(cos(wc
+ wm)t
+ cos(wc
- wm)t)}
The following can also be used:-
v = ac (1 +
m sin wmt)
sin wct = ac{sin wct
+ m/2(cos(wc
+ wm)t
- cos(wc
- wm)t)}
v = ac (1 +
m cos wmt)
sin wct = ac{sin wct
+ m/2(sin(wc
+ wm)t
+ sin(wc
- wm)t)}
v = ac (1 +
m sin wmt)
cos wct = ac{cos wct
+ m/2(sin(wc
+ wm)t
- sin(wc
- wm)t)}
Fn = 1 to 4 shows these 4 ways of describing an AM signal. They
differ in the phase of carrier and modulating signal but the appearance of the AM is the same in all cases.
The applet allows changes in the phase of the carrier and upper
sideband when the phase is changed. Carrier phase changes only affect
the AM when the AM is derived by summing a carrier and 2 sidebands
but not when the AM is derived from multiplying the carrier with
the offset modulating signal.
Fn = 5 shows modulation with 2 sinusoids. The spectrum now has 2 pairs of
sidebands and a total modulation index can be calculated from the 2
modulation indices.
Fn = 6 to 11 shows AM signals with different non sinusoidal
modulating signals. In each case the spectrum and total modulation
index is obtained from the Fourier Series representation of the
modulating signal. Fn = 9 to 11 shows cases where the average of the
modulating signal is zero but the magnitude of the positive peak is
different from the magnitude of the negative peak. In all these
cases the appearance of the AM can be very different from what one
might expect from the total modulation index.
Fn = 8 shows a modulating signal which can be changed by 'a'.
Fn = 12 shows how the AM is built up from the carrier and
sidebands for the case where the spectrum has several sidebands.
The number of sidebands summed can be changed by 'ns'.
Fn = 11,12 show a modulating signal which can be changed by 'b'.
Fn = 13 illustrates the phasor interpretation of the AM. Position
of Phasor plot can be changed by changing Pt. In this plot, the vertical
axis is real while the horizontal axis is imaginary.
The resultant AM moves up and down along the real axis
because the AM does not have any phase variation.
Fn = 14 shows a phasor interpretation of what happens when the
carrier is shifted by 90º. The signal here is obtained by
adding the shifted carrier and the 2 sidebands. Since the resultant has
components along the Vertical and Horizontal axes, then the
resulting signal has both amplitude and phase variations and is not
an AM signal. The orange x shows the amplitude(length of yellow line) variation. The unmodulated carrier is shown as dark red and the difference in the zero crossings between the cyan and
dark red curves is a measure of the phase variation.
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COPYRIGHT © 1996, 2010 Cuthbert Nyack.