By Gabor uncertainty, I mean the principle of uncertainty as applied to signals — with the result that you can't have arbitrary time and frequency localization.
By way of background to my question, I wrote about this recently in a blog post. In the last paragraph, I stated:
Signals do not have arbitrarily precise time and frequency localization. It doesn’t matter how you compute a spectrum, if you want time information, you must pay for it with frequency information.
The point here is that, as I understand it, this uncertainty is an irreducible property of signals. It is not a 'feature' (or bug) in the Fourier transform or mathematics generally. And you can't get around it by using non-Fourier methods like matching pursuit.
I'm now unsure this is really true. Someone wrote to me recently:
...the uncertainty is there because of the math, not nature. A 'signal' is a mathematical construct... The timing information of a signal is in the phase, and for a short-lived pure tone you can exactly recover the timing and frequency from the temporal phase.
I guess I buy this. But when we're measuring unknown, real, broadband, noisy signals does it still hold?