resolving two frequencies less than an fft bin apart?

Humans resolve 2 nearly equal musical tones by listening for the
"beat" notes (given that trigonometry formulas indicate that adding
2 sinusoids is equivalent to modulating one sinusoid).  So why can't
one resolve two very close sinusoids by examining the envelope of
the mixed waveform for some sort of modulation artifact?  If the
envelope fits some fraction of one full cycle of modulation, then
that fraction could be related to the difference between the two
frequencies by less than 1 fft bin.

So I'm wondering what methods could be used to find the best
fit of the envelope of a higher frequency waveform to some
fractional portion of a sinusoid (the very low frequency modulator).


IMHO. YMMV.
-- 
rhn A.T nicholson d.0.t C-o-M

Ron N. wrote:
> Humans resolve 2 nearly equal musical tones by listening for the
> "beat" notes (given that trigonometry formulas indicate that adding
> 2 sinusoids is equivalent to modulating one sinusoid).

Modulating how? Only approximately AM; that has a carrier and two 
sidebands. Only approximately FM. Close enough to either so the ear 
can't tell.

>  So why can't
> one resolve two very close sinusoids by examining the envelope of
> the mixed waveform for some sort of modulation artifact?  If the
> envelope fits some fraction of one full cycle of modulation, then
> that fraction could be related to the difference between the two
> frequencies by less than 1 fft bin.

Provided that the signal is pure and steady enough, and lasts long 
enough to encompass a good part of a beat cycle, it can work just fine. 
But if it lasts that long, the bins will be close enough so there's 
little gained.

> So I'm wondering what methods could be used to find the best
> fit of the envelope of a higher frequency waveform to some
> fractional portion of a sinusoid (the very low frequency modulator).

Yah.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

Ron N. wrote:
> Humans resolve 2 nearly equal musical tones by listening for the
> "beat" notes (given that trigonometry formulas indicate that adding
> 2 sinusoids is equivalent to modulating one sinusoid).  So why can't
> one resolve two very close sinusoids by examining the envelope of
> the mixed waveform for some sort of modulation artifact?  If the
> envelope fits some fraction of one full cycle of modulation, then
> that fraction could be related to the difference between the two
> frequencies by less than 1 fft bin.
>
> So I'm wondering what methods could be used to find the best
> fit of the envelope of a higher frequency waveform to some
> fractional portion of a sinusoid (the very low frequency modulator).
>
>
> IMHO. YMMV.
> --
> rhn A.T nicholson d.0.t C-o-M

Ron, if you do a google search on "Frequency estimation" you'll find
a few good links.

There are methods that have "super-resolution" property, i.e.
resolution
beyond the FFT bin spacing.  Examples include MUSIC, ESPRIT, etc.

This question is asked periodically here:  a search through what other
people have discussed in the past may be good, too.

Hope that helps,
Julius