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Re: human versus spectral resolution



At 7:49 AM +0200 4/3/08, Beerends, J.G. (John) wrote:
A good theory is never wrong, it only has a limited context in which it
is valid.

But a bad theory can be wrong.

The point is not that so much that some theories are wrong, but that all theories are wrong in some sense.


George Box is quoted as saying: "All models are wrong, some are useful". Donald Hebb made a similar point (see my chapter on pitch models in the Springer handbook). In other words: asking if a theory is wrong is the wrong question to ask. We should ask whether it's useful.

With respect to the original question, Nordmark (1968, 1970) made a very useful point. The word 'frequency' is commonly used with two meanings. One is the reciprocal of a time interval, for example the interval between two events in the waveform, or the position along the lag axis of the peak of an autocorrelation function. This, he called 'phase frequency' (Kneser 1948). The other is the locus of a spectral feature determined by Fourier analysis, called 'group frequency'. To quote Nordmark: "For a time function of limited duration, [Fourier] analysis will yield a series of sine and cosine waves grouped around the phase frequency. No exact value can be given [to] the group frequency, which is thus subject to the uncertainty relation."

In practice, the period of a waveform can be determined with arbitrary accuracy from a limited segment of waveform (roughly: 2 periods of the lowest expected fundamental). This can be done by a machine, and possibly also by the auditory system. I don't see a reason to believe that the auditory system would do better than a machine on such simple tasks.

This does not mean that it's not worth looking at how the auditory system does these things. For one thing, knowing how the auditory system does a simple task such as frequency analysis can provide inspiration to help develop algorithms that can then be run on a computer. Those algorithms do not need to be faithful models of the auditory mechanism to be useful. For another, there are more complex tasks such as speech recognition for which we don't know the nature of the problem well enough to get a machine to do it. So I agree wholeheartedly with Dick Lyon's position that there's lots to be learned (from a practical point of view) from studying the auditory system.

Nordmark's point not well known or understood, even by savvy signal processing specialists. It's very common to read that the Heisenberg/Gabor principle (earlier suggested by Helmholtz) limits the accuracy of frequency measurement, and that the fact that one can do better is a paradox.

Alain

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Nordmark JO (1968) Mechanisms of frequency discrimination. J Acoust Soc Am 44:1533-1540.
Nordmark JO (1970) Time and frequency analysis. In Tobias JV (ed) Foundations of modern auditory theory. New York: Academic Press, 55-83.
Kneser (1948). "Bemerkungen über Definition und Messung der Frequenz." Archiv der Elektrishen Übertragung 2: 167-169. [I haven't read this: can anyone confirm if it's relevant?]