From: Charles Henry <czhenry@...>

To: <linux-audio-dev@...>

Subject: Re: [LAD] twice as loud

Date: Tuesday, July 27, 2010 - 8:44 pm

Ah! That's just a bandwidth limitation, but it's a rather goodOn Tue, Jul 27, 2010 at 12:38 PM, Ralf Mardorf

wrote:> It's not impossible. I guess nobody is able to note, let's say, 10 000

example for the mathematically inclined. Let's say we're just talking

about the set of sounds that are 1 second long or less.

I'd like to show that the human auditory system performs a significant

reduction in the dimensionality of sounds. Start with sets of signals

on [0,1] that have finite energy and power: s(t) on [0,1] is finite,

and the integral of s(t)^2*dt on [0,1] is also finite.

Q: So, how many dimensions do we start out with?

A: infinite--this is one example of a Hilbert space. The

dimensionality is clear by application of Fourier series. We can

represent functions in this space with a series of orthogonal

functions (sines and cosines), but to represent *all* functions in

this space, the series has to be infinitely long.

Q: Now suppose we limit the bandwidth to 200 kHz. How many

dimensions do we need?

A: 400,000. By Nyquist's sampling theorem, we need 400,000 samples

to represent continuous signals up to 200 kHz. Either by

sampling/reconstruction or Fourier Series, we can show that our space

is homeomorphic to R^400,000.

So, your own example shows that if we increase our bandwidth

arbitrarily high, we can't tell the difference anymore. The auditory

system is bandwidth limited in this way--typical rule of thumb is

about 20kHz of bandwidth. We represent these continuous sounds with

samples at a rate more than twice the bandwidth. So typically, we

sample at 40kHz and above. Real acoustic sounds can have a lot of

extra frequencies above 20kHz, so sample at higher rates to reduce

aliasing of those frequencies onto the auditory band. No further

increases in quality can be obtained by sampling at faster rates.

Regardless, it's a gigantic number of dimensions. The essence of

psychology is the study of mental representations. How can each of

those things be represented in the mind? The problem becomes, what is

the smallest integer-dimensional space into which we can embed the

space of all sounds? This is not a problem that has been solved, nor

do I prescribe how to take such a measurement.

But finding such a result is the *exact* problem to solve in

psychoacoustic coding. It's reducing a problem from a set which takes

a large number of points to represent all possibilities to a set which

takes the fewest number of them.

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Re: [LAD] twice as loud, Charles Henry, (Tue Jul 27, 8:44 pm)

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