Wavelet Dreams

Paul,

In continuation of your "Uncertainty Principle" thread, I found a few interesting details at:

http://engineering.rowan.edu/~polikar/WAVELETS/WTtutorial.html

From Part 2 (referring to the Fourier Transform):
<< if you are not interested in at what times these frequency components occur, but only interested in what frequency components exist, then FT can be a suitable tool to use.>>

From Part 2 (referring to the Short-term Fourier Transform, ie "real time"):

<<The problem with STFT is the fact whose roots go back to what is known as the Heisenberg Uncertainty Principle . This principle originally applied to the momentum and location of moving particles, can be applied to time-frequency information of a signal. Simply, this principle states that one cannot know the exact time-frequency representation of a signal, i.e., one cannot know what spectral components exist at what instances of times. What one can know are the time intervals in which certain band of frequencies exist, which is a resolution problem.>>

From Part 2 (regarding Time/Frequency, where FT = Fourier Transform):

If we use a window of infinite length, we get the FT, which gives perfect frequency resolution, but no time information. Furthermore, in order to obtain the stationarity, we have to have a short enough window, in which the signal is stationary. The narrower we make the window, the better the time resolution, and better the assumption of stationarity, but poorer the frequency resolution

And, from Part 3 (referring to ALL frequency domain transforms, wavelets included):

<<Although the time and frequency resolution problems are results of a physical phenomenon (the Heisenberg uncertainty principle) and exist regardless of the transform used, it is possible to analyze any signal by using an alternative approach called the multiresolution analysis (MRA) . MRA, as implied by its name, analyzes the signal at different frequencies with different resolutions. Every spectral component is not resolved equally as was the case in the STFT.>>

Yet (From Part 3, continued), Mother Nature exacts a price for such trickery:

<<MRA is designed to give good time resolution and poor frequency resolution at high frequencies and good frequency resolution and poor time resolution at low frequencies.>>

All in all, your "Uncertainty Principle" posit seems to hold for all inorganic spectral processors. Why should the same physics not apply to our nervous systems? Indeed, why tune your strings at all, as long as your melodic style is full of rapid fire glissandos! :) (and you are playing a_sine wave_).

The next step (if there is one) is to factor in the existence of harmonic overtone "clues" and reinforcements of the fundamental pitches involved...

"Certainly", this may bias the pitch recognition process towards a more rapid convergence, no doubt at the price of "transmitting" more spectral energy in order to do so, and also vastly complicating the theoretical terrain, thus leading to a further "uncertainty principle" in anyone being able to follow what another is setting forth conceptually, characterized by a Gaussian amplitude distribution where spectral density per unit cycle is uniform ( = white noise).

Mission complete!

Cheers, J Gill

Why not post this, or just your conclusions, to the tuning list,
under the original topic heading?

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> Why not post this, or just your conclusions, to the tuning list,
> under the original topic heading?

In tuning@yahoo, message #29664, Paul Erlich wrote:

<<I suspect a meaningful modeling of the situation I'm interested in
would have to involve something like a wavelet transform (thanks Dave
Keenan) -- if you know much about these, please respond over at

tuning-math@yahoogroups.com>>

Guess I got confused by your previous statement...

J Gill