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My understanding is that the standard deviation of the Gaussian window in a Gabor filter dictates the temporal resolution.

Wouldn't it always be better then to make the window smaller, thus achieving a high temporal resolution of exactly when in the input signal the frequency of interest (equal to the frequency of the sinusoid in the Gabor Filter) occurs?

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  • $\begingroup$ Sure, if you’re tracking a signal with known and very narrow bandwidth (a sinusoid with fixed frequency being as narrow as it gets). But if you’re interested in a broader bandwidth, then you need the analysis window to be at least long enough to hold a full period of the lower frequency of interest. $\endgroup$
    – Jdip
    Sep 14, 2023 at 16:03

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If your gaussian window width goes to zero, you are, in essence, sampling the signal, $x(t)$, (that was multiplied by the complex exponential, $e^{j 2 \pi f_0 t}$, which always has a magnitude of 1) with a dirac impulse. So the value, after integration, is equal to your instantaneous sample with a known complex phase attached to it.

The output of the Gabor filter will essentially be the output of a one-sided AM modulator.

$$ y(t) = x(t) \cdot e^{j 2 \pi f_0 t} $$

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