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?

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    $\begingroup$ well, since signals occur only in physical reality (is this what you mean by "nature"?), then i would say that the Uncertainty principle exists with any quantitative function that exists in linear time or space. the Fourier Transform is, admittedly, a mathematical construct (so it's an "invention"), but that doesn't matter. whether someone is taking the Fourier Transform of some signal or not, the mathematical reality of it still applies. and what your friend "Someone" wrote is, strictly speaking, incorrect when he/she uses the word "exactly" along with "short-lived". $\endgroup$ – robert bristow-johnson May 26 '14 at 22:25
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    $\begingroup$ 'Nature' was an odd choice of word; 'physical reality' is closer. I guess I meant 'a property of the signal itself', as opposed to our mathematical tools. Maybe even that is philosophically hazy. Anyway, thank you for the comment — I think I follow, and agree with you. $\endgroup$ – kwinkunks May 27 '14 at 0:43
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    $\begingroup$ Signal-to-noise ratio is also a factor in this uncertainty. If you know your signal is a pure exact infinite length sinusoid in zero noise, then 3 or 4 non-aliased samples will give you both exact frequency, magnitude and phase, complete information. Once you add noise (change any one of those 3 or 4 samples), then you need a lot more samples to estimate if something might have changed about either/or any of the sinusoid's properties, or the signal's envelope, or nothing happened. $\endgroup$ – hotpaw2 May 27 '14 at 15:18
  • $\begingroup$ A good point, but noise seems like a whole other issue — but I think time-frequency decompositions are necessarily uncertain, even in the absence of noise, are they not? To put it another way: 'infinite-length sinusoids' don't exist, so cannot be 'pure' or 'exact'. I think. $\endgroup$ – kwinkunks May 28 '14 at 1:10

Time/frequency uncertainty is a property of the meanings or definitions attached to the terms "time" localization and "frequency" estimation. It also alludes to some apriori knowledge, or lack of it, about the characteristics of the signal in question.

  • $\begingroup$ Thank you, but I am not sure I follow. Do you mean that uncertainty is a consequence of having no a priori knowledge of a signal? $\endgroup$ – kwinkunks May 28 '14 at 1:12

Assuming this a a real-world signal, the inevitable presence of noise will to some degree result in uncertainty. Anything beyond the Nyquist of a Fourier transform is not modeled and therefore by definition there is uncertainty about higher frequency components of the signal. For the real world signal, Gabor uncertainty probably has to exist because even if localization in time is possible, there is a limit beyond which localization of phase or frequency is impossible. Separately, as frequency can only be measured within a window of time, there will always be a degree of limitation beyond which only time or frequency can be accurately measured, and not both.

  • $\begingroup$ OK, a good point about signal beyond Nyquist, thank you. That seems like yet another type of uncertainty about the signal. I think Gabor's uncertainty applies strictly to the unaliased signal we can measure. Then, as you say, another problem is that we have to take time chunks to measure frequency in. The chunks are spread across time samples, but finite and therefore spread across frequency samples too. But I think it's even worse than this — the signal itself (regardless of windows) is also spread out but finite in time. $\endgroup$ – kwinkunks May 28 '14 at 1:19

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