The Gabor-Heisenberg uncertainty represents the fundamental limit of time and frequency resolution one can extract from a signal.
$\sigma_t \sigma_f \geq \frac{1}{4\pi}$
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).