# Can the standard deviation of the Gaussian window in a Gabor filter be made infinitesimally small?

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?

• 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.
– Jdip
Sep 14 at 16:03

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