# Linear Filter

#### Statistical Glossary

Linear Filter:

A linear filter is the filter whose output is a linear function of the input. Any output value of a linear filter is the weighted mean of input values. In other words, to form one element of the output at time , it is necessary to multiply the input values for time moments adjacent to by coefficients and to sum up the products.

Mathematically, the output of a linear filter may be described by the expression

where

• is the output of the filter (the result of filtering);
• is the input of the filter (the original time series to be filtered);
• is the size of the “window” of the filter – the number of the input values affecting one output value;
• are “weights” that completely describe any linear filter.

In contrast to nonlinear filter , there is a well developed and conceptually rich mathematical theory of linear filters.

All the linear filters are subdivided into two broad categories – nonrecursive filters and recursive filters .

Examples of a linear filters are: the rectangular filter , the triangular filter , the Gaussian filter , the exponential filter , Kalman filter .

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