[sdiy] Beat tones, Fourier analysis, and nonlinearities in the ear [was Phase shifts and instantaneous frequency]

[Ian Fritz]

At 08:20 PM 7/15/2008, Aaron Lanterman wrote:

>Oh, the ear and auditory system is all sorts of nonlinear. I'm not
>arguing that. I just am unsure about the theory that a 100 Hz tone and
>a 201 Hz tone are multiplying in the ear and creating a 101 Hz spur
>that's beating against the 100 Hz tone, creating the 1 Hz that you
>hear.

This isn't something I made up.  It has been studied extensively, starting 
with Helmholtz's work in the 19th century and continuing up to more recent 
studies where probes were surgically inserted into the ear.


>You can directly _see_ the 1 Hz pattern in the waveform

Of *course* you can see the 1 Hz beats. There is an amplitude difference 
when sinusoids  are added together with different relative phases.  That 
doesn't mean there is a physical signal at 1 Hz.  In linear theory the only 
frequencies that can be present are the original tones and their sum and 
difference frequencies.  None of these is at 1 Hz.  It doesn't matter a bit 
what your eye sees.  The 1 Hz detuning is simply a convenient quasi-static 
method to demonstrate the perceived timbre change.  You would hear the same 
timbre change if the signals were at 100 Hz and 200.0001 Hz.  The 
comparison would be difficult because the change would be so gradual.  You 
can also hear the same timbre change if you record a superpositioned 100 Hz 
and 200 Hz signal with two different phases and make an A/B comparison. I 
guess nobody is going to understand this unless I do it for you.


>Consider playing a 439 Hz tone against a 441 Hz tone. Your ear
>perceives a 440 Hz tone modulated by a 1 Hz beat, so you'll hear two
>peaks per second, and if you plot a _short time_ Fourier transform of
>it with a small window, that's what you'll see. You'll see it in the
>spectrogram and you'll hear it with your ears, and you can describe
>that without ever having to multiply cos(439 t) with cos(441 t).

But in this case 2 Hz is the difference frequency of the two signals, so of 
*course* you see it in the spectrum and there is energy at that 
frequency.  But this is a qualitatively different situation.  Once again, 
with 100 Hz and 201 Hz there is *no* low frequency difference frequency.


>I think maybe an important distinction to make is about short-term vs.
>non-short term Fourier analysis.

Why?  We are only considering (quasi) steady tones.  But go ahead and do a 
short-time analysis of the 100 Hz + 201 Hz signal.  There won't be any 
significant change in the spectrum with time.  And if the small change 
there is bothers you, redo everything with 100 Hz and 200.1 Hz or 200.00001 
Hz.  You will hear the same timbre change.


>You could write down an equation for an exponentially damped sinusoid
>x(t). LIstening to it, you'll hear a tone that gradually decreases in
>volume.
>
>You could then write down its Fourier transform, which would look like
>two copies of 1/(a+j w) centered at +/- the frequency of your
>sinusoid. In this representation, time's been collapsed, and you
>imagine the wave is made up of the continuum of frequencies - in fact,
>there'd be frequency content at ALL frequencies. But you don't
>perceive all those frequencies.

Because they are very weak and masked by the main signal.  If the envelope 
is short enough that the various frequency components are strong enough to 
be physically perceived then you *do* hear them.  There is a distinct click 
or pop (as I'm sure everyone knows from using EGs with short A/D times).


>Neither the time-domain or frequency-domain equation fully describe
>what we perceive. For that, a short-time Fourier analysis is best.

Again, the signals are very slowly varying (quasi static). I don't see any 
need for short-time Fourier analysis.

   Ian