Newsgroups: digex.general
From: esmythe@digex.net (Erich Smythe)
Subject: QAM Primer (was Re: Baud, BPS, etc.)

The most common form of high speed modulation over voice telephone lines
is Quadrature Amplitude Modulation (QAM).  What is that, you ask?  QAM
sends information by modulating the phase and amplitude of a carrier
wave.  That was clear, wasn't it?  Well, this is best described using a
few simple objects found in nearly every modern home.  Allow Dr. Nerd to
explain: 

First take a sheet of paper, lay it lengthwise, and draw a horizontal axis
roughly in the middle of the page.   That is our time axis.  Now draw a
sine wave with several cycles above and below the axis proceeding from
left to right (or right to left if your native writing system does that). 
That is our carrier wave.  For those that found trigonometry boring, a
sine wave is a curvy wave that rises above an axis to a point and then
below the axis the same distance below the axis.  It is symmetrical above
and below the axis.  The maximum height of the wave is the Amplitude.  The
wave completes a Cycle when it goes above and below the axis and returns
to the same amplitude value.  The Phase of the wave is roughly where you
are in the cycle.  The number of cycles in a second is called the
frequency.

Now lets modulate.  Take a reasonably clear condom, unlubricated and
unused, and without all those funny ribs and tickler thingies.  Unroll the
condom.  Cut the condom lengthwise (opposite the "Bobbit Cut") and lay it
flat over your sine wave.  WARNING:  This action renders the condom
useless for its intended purpose!  Trace the sinewave on the condom using
a felt tip pen.

Shift the condom along the horizontal axis.  This is called Phase Shift
Keying (PSK) since you have shifted the phase.  Stretch the condom in the
vertical axis.  This is called Amplitude Shift Keying (ASK).  Do PSK and
ASK simultaneously and you have QAM.  This is how modems communciate.  Now
stretch the condom along the horizontal axis.  You have enlarged the
wavelength (men, stop dreaming) and have in effect shifted the frequency. 
This is called Frequency Shift Keying (FSK) and has nothing to do with QAM
but I just felt like throwing it in.

Phase shifts are measured in degress (or radians), so that a shift of half
a cycle is 180 degress, 3/4ths of a cycles is 270 degrees, and a full
cycle is 360 or zero degrees.  You can define a QAM Constellation using a
phase-amplitude plot.  The phase is plotted like a compass, with a ray
pointing from the center of the axes.  The amplitude is the length of the
ray.  A modem constellation is a bunch of phase and amplitude points, and
for V.32 it is a sqare centered at the orgin.  At every baud the phase and
amplitude are measured and matched with a constellation point.  This is
called a symbol.  For V.32 without a trellis (9,600 bps) this is done 2400
times a second: the baud rate.

How do we get more bits than bauds?  For V.32 there are 16 constellation
points that encode 4 bits for each symbol.  So there are four bits
transmitted with every baud.  4 bits times 2400 cycles/sec is 9600
bits/sec, our data rate.  V.32bis at 14,400 actually uses a 128 point
constellation with 7 bits per point, but because it uses a special
encoding called a trellis encoding that eats a bit in each symbol you get
six bits/symbol, or 14,400 bps.  Trellis codes give Dr. Nerd a headache,
and he will not explain those today.  V.34 (28,800) uses a 916 point
constellation and a four-dimensional trellis code, giving Dr. Nerd four
times the headache so he won't explain that one, either.

Simple, wasn't it?

-erich