Gaussian filter

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Linear analog electronic filters
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In electronics and signal processing, a Gaussian filter is a filter whose filter window is the Gaussian function

$x \mapsto \sqrt{\frac{a}{\pi}}\cdot e^{-a \cdot x^2}$

or with the standard deviation as parameter

$x \mapsto \frac{1}{\sqrt{2\cdot\pi}\cdot\sigma}\cdot e^{-\frac{x^2}{2\cdot\sigma^2}}$ .

Mathematically, a Gaussian filter modifies the input signal by convolution with a Gaussian function; this transformation is also known as the Weierstrass transform.

Gaussian filters are designed to give no overshoot to a step function input while minimizing the rise and fall time (which leads to the steepest possible slope). This behavior is closely connected to the fact that the Gaussian filter has the minimum possible group delay.

[edit] Digital implementation

Since the Gaussian function decays rapidly, it is reasonable to truncate the filter window and implement the filter directly for narrow windows.
Since the Fourier transform of the Gaussian function yields a Gaussian function, again, you can apply the Fast Fourier transform to the signal (preferably divided into overlapping windowed blocks), multiply with a Gaussian function and transform back. This is the standard procedure of applying an arbitrary finite impulse response filter, with the only difference that the Fourier transform of the filter window is explicitly known.
Due to the central limit theorem you can approximate the Gaussian by several runs of a very simple filter like the moving average. The simple moving average corresponds to convolution with the constant B-spline, and e.g. four iterations of a moving average yields a cubic B-spline as filter window which approximates the Gaussian quite well. You can interpret the standard deviation of a filter window as a measure of its size. For standard deviation $σ$ and sample rate $f$ you obtain the frequency $\frac{f}{\sigma}$ which can be considered the cut-off frequency. A simple moving average corresponds to a uniform probability distribution and thus its filter window with size $n$ has standard deviation $\sqrt{\frac{n^2-1}{12}}$ . Thus $m$ moving averages with sizes $n_1,\dots,n_m$ yield a standard deviation of $\sqrt{\frac{n_1^2+\dots+n_m^2-m}{12}}$ . (Note that standard deviations do not sum up, but variances do.)