Sinusoidal Amplitude Modulation (AM)
It is instructive to study the
modulation of one
sinusoid by
another. In this section, we will look at sinusoidal
Amplitude
Modulation (AM). The general AM formula is given by

where

are parameters of the sinusoidal
carrier wave,
![$ \alpha\in[0,1]$](http://www.dsprelated.com/josimages_new/mdft/img482.png)
is called the
modulation index (or
AM index),
and
![$ a_m(t)\in[-1,1]$](http://www.dsprelated.com/josimages_new/mdft/img483.png)
is the
amplitude modulation signal. In
AM radio broadcasts,

is the audio signal being transmitted
(usually bandlimited to less than 10 kHz), and

is the channel
center frequency that one dials up on a radio receiver.
The modulated signal

can be written as the sum of the
unmodulated carrier wave plus the product of the carrier wave and the
modulating wave:
 |
(4.1) |
In the case of
sinusoidal AM, we have
 |
(4.2) |
Periodic amplitude modulation of this nature is often called the
tremolo effect when

or so (

Hz).
Let's analyze the second term of Eq.

(
4.1) for the case of sinusoidal
AM with

and

:
 |
(4.3) |
An example waveform is shown in Fig.
4.11 for

Hz and

Hz. Such a signal may be produced on an analog synthesizer
by feeding two differently tuned
sinusoids to a
ring modulator,
which is simply a ``four-quadrant multiplier'' for analog signals.
Figure:
Sinusoidal amplitude modulation as in Eq.
(4.3)--time
waveform.
![\includegraphics[width=3.5in]{eps/sineamtd}](http://www.dsprelated.com/josimages_new/mdft/img494.png) |
When

is small (say less than

radians per second, or
10 Hz), the signal

is heard as a ``beating
sine wave'' with

beats per second.
The beat rate is
twice the modulation frequency because both the positive and negative
peaks of the modulating sinusoid cause an ``amplitude swell'' in

. (One
period of modulation--

seconds--is shown in
Fig.
4.11.) The sign inversion during the negative peaks is not
normally audible.
Recall the trigonometric identity for a sum of angles:
Subtracting this from

leads to the identity
Setting

and

gives us an alternate form
for our ``ring-modulator output signal'':
![$\displaystyle x_m(t) \isdef \sin(\omega_m t)\sin(\omega_c t) = \frac{\cos[(\omega_m-\omega_c)t] - \cos[(\omega_m+\omega_c)t]}{2} \protect$](http://www.dsprelated.com/josimages_new/mdft/img505.png) |
(4.4) |
These two sinusoidal components at the
sum and difference
frequencies of the modulator and carrier are called
side bands
of the carrier wave at frequency

(since typically

).
Equation (
4.3) expresses

as a ``beating sinusoid'', while
Eq.

(
4.4) expresses as it two
unmodulated sinusoids at
frequencies

. Which case do we hear?
It turns out we hear

as two separate tones (Eq.

(
4.4))
whenever the side bands are
resolved by the ear. As
mentioned in §
4.1.2,
the ear performs a ``short time
Fourier analysis'' of incoming sound
(the
basilar membrane in the
cochlea acts as a mechanical
filter bank). The
resolution of this
filterbank--its ability to discern two
separate spectral peaks for two sinusoids closely spaced in
frequency--is determined by the
critical bandwidth of hearing
[
45,
76,
87]. A critical
bandwidth is roughly 15-20% of the band's center-frequency, over most
of the audio range [
71]. Thus, the side bands in
sinusoidal AM are heard as separate tones when they are both in the
audio range and separated by at least one critical bandwidth. When
they are well inside the same
critical band, ``beating'' is heard. In
between these extremes, near separation by a critical-band, the
sensation is often described as ``roughness'' [
29].
Equation (
4.4) can be used to write down the spectral representation of

by inspection, as shown in Fig.
4.12. In the example
of Fig.
4.12, we have

Hz and

Hz,
where, as always,

. For comparison, the spectral
magnitude of an
unmodulated 
Hz
sinusoid is shown in
Fig.
4.6. Note in Fig.
4.12 how each of the two
sinusoidal components at

Hz have been ``split'' into two
``side bands'', one

Hz higher and the other

Hz lower, that
is,

. Note also how the
amplitude of the split component is divided equally among its
two side bands.
figure[htbp]
Recall that

was defined as the
second term of
Eq.

(
4.1). The first term is simply the original unmodulated
signal. Therefore, we have effectively been considering AM with a
``very large'' modulation index. In the more general case of
Eq.

(
4.1) with

given by Eq.

(
4.2), the magnitude of
the spectral representation appears as shown in Fig.
4.13.
figure[htbp]
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