On Mon, 16 Jul 2007 12:28:07 -0700, Jim Kelley
wrote:

John Fields wrote:

On Fri, 06 Jul 2007 19:04:00 -0000, Jim Kelley
wrote:

In your example, with 300Hz and 400Hz as the carriers, the sidebands
would be located at:

f3 = f1 + f2 = 300Hz + 400Hz = 700Hz

and

f4 = f2 - f1 = 400Hz - 300Hz = 100Hz


both of which are clearly within the range of frequencies to which
the human ear responds.

Indeed. We would hear f3 and f4 if they were in fact there.

Your use of the term "beat frequency" is confusing since it's
usually used to describe the products of heterodyning, not the
audible warble caused by the vector addition of signals close to
unison.

The term is commonly used in describing the results of interference in
time, as well as for mixing.

Since the response of the ear is non-linear in amplitude it has no
choice _but_ to be a mixer and create sidebands.

Perhaps you're confusing log(sin(a)+sin(b)) with
log(sin(a))+log(sin(b)).

---
Perhaps, but I don't think either of those is correct, since for
mixing to occur (AIUI, for sidebands to be generated) the sine waves
themselves must be multiplied at the lowest level of the equation
instead of added.

That is, the solution of


log(sin(a)+sin(b))


will describe the numerical value of the logarithm of the vector sum
of two sine waves, and since the addition created no sidebands, the
output of the circuitry providing the logarithmic transfer function
will only be the instantaneous value of the logarithm of the vector
sum of the amplitudes of both signals.

Similarly,


log(sin(a))+log(sin(b))


describes the addition of the logarithm of the amplitude of sin(a)
to the logarithm of the amplitude of sin(b), which still produces
only a sum.

That is, no sidebands.
---

If you don't mind me asking, where did you get this notion about the
ears creating sidebands?

---
Well, whether I mind or not it seems you've asked anyway, so your
concern for my sensitivity is feigned.

That, coupled with your relegating it to being a "notion", seems to
be designed to discredit the hypothesis, offhandedly, and make me
work against a headwind in order to prove it valid, with you being
the negative authoritarian blowhard detractor.

If you're really interested in the subject I'll be happy to discuss
it with you if you can keep your end of the discussion objective and
free from pejorative comments.

Otherwise, **** off. ;^) -- Note tongue-in-cheek smiley, :-)


--
JF