On generalizing Gabor-Heisenberg uncertainty in a noisy environment

Question

The Gabor-Heisenberg uncertainty represents the fundamental limit of time and frequency resolution one can extract from a signal.

$sigma_t sigma_f geq frac{1}{4pi}$

My question is: what happen in a noisy environment?

If a bird sings a perfect constant pitch. Depending of the ambient noise, the frequency estimation of this pitch will be affected by the SNR.

Let say you have a time-varying arbitrary tune (like a bird song) played in a arbitrary colored noise environment. Is there a generalized Gabor uncertainty that takes to account the SNR?

I expected something like this:

$sigma_t sigma_f geq function(SNR)$

Given the downvote(s) and the comments, I'll try to rephrase my question:

Frequency estimation is limited by SNR in a noisy environnent.
Frequency estimation is limited by Gabor-uncertainty.

If statements 1 and 2 are correct, how to reconcile the two statements in one general concept? I suppose, there should be a way to compute the 'frequency-estimation precision' as a function of both SNR and time resolution (maybe with some other assumptions).

Cedron Dawg · Answer

This uncertainty thing is often misapplied.  Your statement for #2 is such a misapplication.  The bell curve, aka Gaussian, is an eigenfunction of the FT, (only an approximation in the DFT, but that is a different story).  The uncertainty principle is saying when a shape is taken back and forth from the FT and the narrowest cluster of information is desired, this is accomplished by the Gaussian.  It has nothing to do with parameter estimation, but noise indeed does.

You are conflating different concepts.

My latest blog article gives the actual code for calculating the frequency, amplitude, and phase of a pure tone exactly in the single noiseless pure tone case.  It is also robust in the presence of noise.

A Two Bin Solution

Early on, when trying to show "experts" that I had found exact formula frequencies for tones in the DFT, I was often quoted what you said and they would refuse to even look at my math as if I were proposing a perpetual motion machine.

So many misconceptions exist about the DFT, even among supposed learned folks, and that is scary.

Don't believe me?

Here is an actual quote from an expert who shall remain unnamed.

----------------------------------------------------------
An "exact" computation of frequency based on FTs is not possible for
some very simple reasons:  The Fourier Transform is itself an
estimator, and the time-frequency uncertainty principle means
frequency can't be determined exactly without an infinitely long
observation window.  So essentially all frequency computations of time
series or signals are estimates, it's just that some are more accurate
or more efficient than others.

So claiming that you have an exact solution just isn't going to get
the attention of everyone interested in the field.
----------------------------------------------------------

The only truth in there is the last sentence.  So, I wrote it up in a blog where the math is indisputable and anyone who wants to can verify it for themselves.

No, I've never gotten an apology.

On generalizing Gabor-Heisenberg uncertainty in a noisy environment

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