[sdiy] Phase shifts and instantaneous frequency

[Aaron Lanterman]

On Jul 15, 2008, at 6:19 PM, Ian Fritz wrote:

>> http://www.electricdruid.com/beatwave.jpg
>>
>> Since I can see a 1Hz effect in the waveform in front of me, I'm not
>> entirely convinced by your explanation that this is only a non-linear
>> effect of our hearing. It looks to me like my ears are picking up a
>> 1Hz variation which exists in this waveform.
>
> The phase difference will, indeed, produce a different summed  
> waveform.  But the power spectrum is unchanging, i.e. there is  
> always the same power in each component.  So if the ear's response  
> is linear you should hear a steady tone.  This is the usual argument  
> that is used to claim that relative phase shouldn't matter, even  
> though the waveform itself changes.

I'm wondering though if this is really best described as an  
intermodulation (i.e. multiplication) effect happening in the auditory  
system?

If I imagine that somewhere in my auditory system, a 100 Hz wave and a  
201 Hz wave are being multiplied, then I get sum and difference tones  
of 301 Hz and 1 Hz. I don't think anyone is perceiving a 301 Hz tone,  
but you're not really picking up on a 1 Hz tone either.

Another way to look at the signal would be as a wave with a 1 Hz  
fundamental (missing) and a 100th harmonic and 201th harmonic. You'd  
formally write the fourier series like that, but of course that's now  
how you perceive the tone. The frequency-sensing aspect of your brain  
wants to hear a 100 Hz tone with a first and second harmonic, but the  
time-sensing part of your brain is hearing that 1 Hz variation, since  
the wave is periodic with a period of 1 second. Your perception of  
that one second time variation may change at different volumes or with  
different stereo spreads.

Any sum of two tones could be mathematically written as a  
multiplication of a single tone in between the two frequencies a beat.

cos(a) + cos(b) = 2 cos((a+b)/2) cos(a-b/2)

If a and b are far apart, you perceive things as on the left; if  
close, you perceive as on the right.  This can be graphed by plotting  
a spectrogram with different window sizes. If you use a big size, you  
have high frequency resolution, and low time resolution, so you see  
two lines, as indicated on the left. If you use a small window size,  
you have good time resolution, but poor frequency resolution, so you  
see a single line that fades in and out.

- Aaron