Definition
The one-dimensional Gaussian filter has an impulse response given by
and the frequency response is given by (eq.206 Fourier transform)
with the ordinary frequency. These equations can also be expressed with the standard deviation as parameter
and the frequency response is given by
By writing as a function of with the two equations for and as a function of with the two equations for it can be shown that the product of the standard deviation and the standard deviation in the frequency domain is given by
- ,
where the standard deviations are expressed in their physical units, e.g. in the case of time and frequency in seconds and Hertz.
In two dimensions, it is the product of two such Gaussians, one per direction:
where x is the distance from the origin in the horizontal axis, y is the distance from the origin in the vertical axis, and σ is the standard deviation of the Gaussian distribution.
Read more about this topic: Gaussian Filter
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