[LAD] PS: twice as loud

On Tue, Jul 27, 2010 at 12:38 PM, Ralf Mardorf <ralf.mardorf(a)alice-dsl.net> wrote: Ah! That's just a bandwidth limitation, but it's a rather good 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.