[bit OT] Heisenberg and digital audio

[Michael Gogins]

Denis Gabor, invented holography and won the Nobel prize, is the author of a
number of seminal papers on what is now called the Gabor transform, which
along with the Fourier transform and wavelets forms the basis of
contemporary signal processing theory.

Gabor pointed out that sound can be represented as a sequence of time
samples (regular digital audio recording), a sequence of frequency samples
(one FFT frame), or as a grid consisting of time on one dimension and
frequency on the other dimension (now called the Gabor transform or Gabor
representation). This can be approximated by the windowed, overlapping,
short-time Fourier transform that is one of the main ways we do things in
time/frequency DSP today. The point is, the size of the cell in the grid is
called the Gabor logon and represents the minimum uncertainty regarding our
knowledge of how much of the acoustical energy in a signal is located at a
particular time and frequency. It is EXACTLY the same mathematics and the
same concept as the Heisenberg uncertainty relation between position and
momentum: the more precisely one knows time, the less precisely one knows
frequency, until with a delta transform (or regular old digital audio
soundfile) each frequency bin contains all frequencies (is a click or delta
impulse); conversely the more precisely one knows frequency, the less
precisely one knows time, until with the classical Fourier transform each
time bin contains all times (is a simple hum containing an unvarying complex
sinusoid).

As a side effect of Gabor's work, we know pretty precisely just how much of
the available physical information in sound the human ear succeeds in
detecting: about half, particularly in the music band, making the ear
perhaps our most acute sense, and giving acoustical engineers and equipment
builders a hell of a hard time making equipment whose defects we can't hear.
In other words, in a poetic sense, the ear may be closer to physical reality
than any other organ.

Do a Web search on the Gabor transform or just the name Gabor, and you'll
get started with references.

-----Original Message-----
From: owner-music-dsp at shoko.calarts.edu
[mailto:owner-music-dsp at shoko.calarts.edu]On Behalf Of Bram De Jong
Sent: Tuesday, September 14, 1999 3:14 PM
To: music-dsp at shoko.calarts.edu
Subject: [bit OT] Heisenberg and digital audio


Hi,



I've been studying rather large quantities of Quantum physics lately (I
took the exam on monday and it was OK :)))).

And, I was thinking about digital audio<->Heisenberg uncertainty
principles.

Has there been done work on this?
I know of the time-freq problems with FFT's, but other than that I've
never seen anything about this subject.
And, I guess there could be equal theorems about the quantization of
energy in the audio area. As audio is waves too.

Anyway: if anyone has ANYTHING usefull to say about this, I'd be very
happy.



Bram



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