The Gaussian filter is a linear filter that is usually used as a smoother . The output of the gaussian filter at the moment is the weighted mean of the input values, and the weights are defined by formula

• is the "distance" in time from the current moment;

• is the parameter of the Gaussian filter;

• is the normalization constant chosen to make the sum of all weights equal to the unit value.

If you plot the values of against , then the plot coincides with the famous bell-like curve describing the density of the gaussian distribution . This explains the word "gaussian" in the name of the filter.

The gaussian filter is completely defined by a single parameter . The greater the value of , the wider the window function , and, hence, the greater the degree of smoothing.

The gaussian filter provides better suppression of higher frequencies than the rectangular filter and the triangular filter .

Besides the one-dimensional gaussian filter described above, there are extensions to the case of two dimensions, say, . Such two-dimensional gaussian filters are widely used in image processing .

See also the short course Time Series Forecasting .