[MistrDave]

Threads like this are a joy for me. Intelligent discussion, and the audio files certainly provide example. I'm often one to say it is important to understand the science, but even more important to experience real world application.
 
After we consider environment, source, equipment and even our own auditory limitations, it truly becomes an analysis of ideals rather than reality. Being able to A/B audio files like these through various levels of equipment makes the laws of diminishing returns ever apparent. 
 
It focuses the attention towards best reproducing what can be heard instead of believing there is something we are "missing".

 

[ultrabike]

Quote:

 
All three waveforms should look the same in continuous time with an "ideal" converter. The fact that they do not on the picture is because of the limitations of the audio editor; it does use sinc interpolation to display the waveform, but it uses only a few neighboring samples, so the reconstruction is not quite perfect.

Thanks stv014, that explains the differences.
 
Increasing the sampling rate might make the number of samples in the interpolation filter less of an issue though 

 

[eucariote]

Quote:

I don't think the brain process instantaneous times of arrival. My understanding is that the brain needs quite a few ms of audio data from both ears to figure out localization.
 

 
It is close to instantaneous.  The superior olivary complex is arranged in layers so that phase differences are detected over many frequencies at once, to overcome ambiguity of multiples of a phase difference over a common frequency.  It can be done over a couple of cycles for the lower frequencies with the majority of neurons coding ~300-2000 Hz, something like 5-10 ms.  BTW this is also why it is so hard to localize pure tones- those electric carts at airports almost always use single tones to warn people, which is the least informative sound possible.
 
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The ideal continuous time representation of a discrete time impulse (one sample of 1.0 at a certain time, and zeros everywhere else) is sin(x) / x, where x is the time difference relative to the time of the impulse, in samples multiplied by ℼ. At x = 0, the value of the function is 1. Obviously, this function extends infinitely in both directions, so a real world approximation needs to be windowed to a finite length; this causes frequency response errors near the Nyquist frequency, and imaging above that, but still does not limit the "resolution" of the phase or frequency to discrete steps. To fully reconstruct the continuous time signal mathematically, you multiply (convolve) each sample value with the above described sinc function, and then sum all the multiplied and sample-delayed sinc functions. This will, for example, turn a coarse, "stair-stepped" digital sine wave into a smooth continuous one. Also, if you offset x by any fraction of ℼ, that will delay the signal by fractions of a sample.

 
Thanks for that- nice to have a mathematician in the house.  So a reconstruction filter can fill in the necessary phase information.  My more general point might not apply anyways in the sense that l/r microphones are probably almost never a similar distance as our ears.  This is something that sound engineers should consider, physiology can be instructive on the relevant information utilized in human auditory processing.

 

[ultrabike]

Quote:

 
It is close to instantaneous.  The superior olivary complex is arranged in layers so that phase differences are detected over many frequencies at once, to overcome ambiguity of multiples of a phase difference over a common frequency.  It can be done over a couple of cycles for the lower frequencies with the majority of neurons coding ~300-2000 Hz, something like 5-10 ms.  BTW this is also why it is so hard to localize pure tones- those electric carts at airports almost always use single tones to warn people, which is the least informative sound possible.
 

 
I think it is more like 150 ms according to this source, but I may be reading it wrong, and that is definitively not my field. Will 5-10 ms give the brain enough to resolve times of arrival in the order of 10-20 us?
 
Quote:

 
Thanks for that- nice to have a mathematician in the house.  So a reconstruction filter can fill in the necessary phase information.  My more general point might not apply anyways in the sense that l/r microphones are probably almost never a similar distance as our ears.  This is something that sound engineers should consider, physiology can be instructive on the relevant information utilized in human auditory processing.

 
I don't think stv014 (and xnor) are mathematicians. I may be wrong, but I think they are Electrical Engineers with some focus on DSP... And so am I...
 
I also think your point about the mics is a valid one. I love binaural recordings