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From: myers@hpfcdj.HP.COM (Bob Myers)
Newsgroups: sci.electronics
Subject: Re: ..but what about _output_ filtering for D/A's?
Message-ID: <17660124@hpfcdj.HP.COM>
Date: 22 Oct 90 19:03:54 GMT
References: <1319@beguine.UUCP>
Organization: Hewlett Packard -- Fort Collins, CO
Lines: 59

>An some people have pointed out, the whole trick to getting the beats
>is a non-linear addition of the two sine waves.

Still "no."  Non-linear ADDITION (whatever that means) is not the source
of true intermodulation.  (By "non-linear addition", I can only assume that
you mean the two input signals are multiplied by different values before being
summed.  But this is not required to produce *interference*, which is the
phenomenon which gives audible "beats" and visible interference patterns -
interference happens when adding any two signals of different phase or
frequency.)

Non-linear *multiplication* (i.e., amplification) is the means through which
electronic equipment can produce the sum and difference frequencies of two
inputs.  Here's what happens:

Ideally, an amplifier would produce and output which is simply a "bigger copy"
of the input; this would equate to a linear function as follows (in terms
of input and output voltages):

    Eout = A Ein

where A is the amplification factor; if the output is a "ten times bigger"
version of the input, A = 10.  As this is of the same form as the equation
for a line, such amplification is said to be "linear," and the gain does
not depend on the input amplitude.

However, NO electronic device can provide perfectly linear behavior over all
possible inputs; at some point, at least, the output will no longer be
properly described by such a simple equation.  If there is any non-linearity
in the output function, then there must be at least some "higher-order"
terms in this function.  We might imagine that a simple case would be at
least:

   Eout = AEin + BEin^2

Here, the "linear" portion of the action (the amplification that we really
want) is described by the first term, and the non-linearity is modelled as
a "squared" term.  Here's where the intermodulation comes from, for if the
input is anything other than a simple sinusoid, there will be at least
two sinusoidal components.  In the simplest case, there would be exactly
two:

   Ein = Xsin(w1t) + Ysin(w2t)

where w1 and w2 are indicating different frequencies.  It should be clear that
if THIS value of "Ein" is squared (by the non-linearity of the amplifier),
then the output will include some terms which are the product of these two
sinusoids (plus the multiplying factor B).  Trig identities will show that
the multiple of two sine waves are cosines at the sum and difference
frequencies, and so we have the "intermodulation" components.

(This has some practical application; for example, a diode is about the
simplest totally-nonlinear device one can imagine, and forms the basis -
along with a resistive summing network - for a simple AM modulator or mixer.)


Bob Myers  KC0EW   HP Graphics Tech. Div.|  Opinions expressed here are not
                   Ft. Collins, Colorado |  those of my employer or any other
myers@fc.hp.com                          |  sentient life-form on this planet.