I do have (I hope!) a simple question. Let's say I have a time series with a discretization of "$dt$" (in real world: yearly values in my specific case). Now I use a Gaussian Filter with $\sigma = 4$, which gives me a half-width of about ~ $6*dt$ (i.e. the ratio of weights $\frac{W_{i}}{W_{i+6}}\approx2)$.
Which time scales are effectively smoothed? My aim is to smooth at least all time scales under 3 time steps (i.e. 3 years).
edit - Latex formatting
The Fourier transofrm of a Gaussian is again a Gaussian - with reciprocal $\sigma$. This means that signals with frequency $\omega$ get multiplied with a factor of $e^{-\omega^2\sigma/2}$. "Under 3 time steps" corresponds to $\omega\ge \frac{2\pi}{3}$, hence your choice of $\sigma=4$ leads to a factor $\le e^{-8\pi^2/9}\approx0.00015$. Already $\sigma=2$ would lead to $\le e^{-4\pi^2/9}\approx0.012$. On the other hand, any frequencies $>\frac{2\pi}2$ are alreday lost in discretization ...
