Modulation and sidebands - Matlab audio demo

1. Introduction and problem statement

When a carrier wave of some frequency fc is modulated by another wave
of frequency fm, SIDEBANDS appear, i.e. other frequencies besides fc
show up in the signal spectrum.

In the case of Amplitude Modulation (AM), sidebands appear at
frequencies fc+fm and fc-fm along with the original carrier wave at
fc.

At first, this may appear confusing, since the carrier wave simply goes
up and down in amplitude, and no new frequencies are added.   When an
audio carrier at say fc=256 Hz is modulated at say fm=2 Hz, then the
modulated wave sounds like a vibrato (warble) on the carrier.  However,
when fm is increased to 20 Hz or so, warble is very fast, and when fm
goes above 50 Hz or so, other tones clearly appear in the sound.

For any modulating frequency fm, the AM signal contains components at
fc, fc+fm and fc-fm.  If fm is less than 20 Hz or so, then the human
ear perceives the modulated signal as a single tone with a warble.  If
fm is more than 50 Hz, then the human ear perceives the separate tones
at fc, fc+fm and fc-fm.

If fm=64 Hz, then we have 3 tones: fc-fm=192 Hz, fc= 256 Hz and
fc+fm=320 Hz.  These are in the ratio 3:4:5, and correspond to musical
notes G, C, E and form a chord.

2. Matlab code

The Matlab code implements the modulated signal as per the equation.
The Matlab code is set up to gradually increase one parameter
(modulating frequency fm or the modulation index ka=beta) from near
zero to some maximum value less than fc.  The Matlab code outputs
audio, and writes a .wav file for later playback.  The Matlab code also
outputs a 3-D plot of the amplitude spectrum and how it changes with
time (one spectrum for each block of 1024 samples).  Later versions may
have 2-D moving video instead.  To run the code with different
parameters, edit the source code file.  Later versions may have a GUI
interface.

3. Audio/visual demos

For AM, when fm is varied, and ka=beta is a constant < 1, the output
illustrates how the tone at fc starts with a slow warble, and then
splits into 3 tones, retaining the original tone at fc, and adding the
two new ones at fc+fm and fc=fm.

For AM, when ka=beta is varied and increased above 1 (i.e.
overmodulation) with constant fm, the output illustrates how the two
sidebands at fc+fm and fc-fm dominate and the carrier remains
constant.  For very large ka=beta, this overmodulated AM is equivalent
to ring modulation.

For double sideband suppressed carrier (DSB) (ring modulation), the
audio illustrates how the single tone at fc splits into two tones at
fc+fm and fc-fm, and the original tone disappears.

For single sideband (SSB), there is only one tone at fc+fm for USB (or
fc-fm for LSB).

For FM, if fm is constant and beta is varied, then more sidebands
appear at fc+-n*fm and vary in amplitude along with beta.  If beta is
constant and fm is varied, then the sidebands are fixed in amplitude
and vary with fm.