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

[cheater cheater]

It's actually 0.5Hz.
Look at the picture carefully.

On 7/16/08, Aaron Lanterman <lanterma at ece.gatech.edu> wrote:
> On Jul 15, 2008, at 9:04 PM, Ian Fritz wrote:
>
>
> > Again, the power spectrum is constant in time.  So doesn't linear signal
> theory say you would not hear beats?  That's the first-order picture, that
> the ear does a linear Fourier analysis.  And if you listen at a low volume
> level you don't hear the beats, whereas they are very pronounces at high
> volume levels.  How do you explain that with a linear theory?  *Any* linear
> theory?
> >
>
>  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. That might be happening,
> or something else might be happening. You can directly _see_ the 1 Hz
> pattern in the waveform without having to go into your synthesis program and
> multiply anything with anything else to simulate that _particular_
> nonlinearity that would give you 101 Hz spur. You can see it without
> invoking any models of the ear at all. Of course, there, we're invoking some
> models of the eye. ;)
>
>  The ear may be perceiving that 1 Hz signal in all sorts of nonlinear ways,
> explaining the changes you perceive at different volumes. For instance, I
> could easily imagine it as popping out of some sort of envelope detection of
> a short-time Fourier transform.
>
>  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).
>
>  I think maybe an important distinction to make is about short-term vs.
> non-short term Fourier analysis.
>
>  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.
>
>  Neither the time-domain or frequency-domain equation fully describe what we
> perceive. For that, a short-time Fourier analysis is best.
>
>  I also could be missing Ian's point...
>
>  - Aaron
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