**Exponential Filter:**

The exponential filter is the simplest linear recursive filter . Exponential filters are widely used in time series analysis , especially for forecasting time series (see the short course Time Series Forecasting ).

The exponential filter is described by the following expression:

where

- is the output of the filter at time moment ;
- is the output of the filter at the previous time moment ;
- is the input of the filter;
- is the parameter of the filter.

In simple words, the output of the exponential filter is the weighted sum of the previous output (taken with weight ) and the current input value (taken with weight ). The smaller the parameter , the longer the "memory" of the exponential filter and the greater the degree of smoothing .

The term "exponential" stems from the fact that, if to try to realize an equivalent nonrecursive filter , then the weights , defining the contribution of the input values to the output , decline exponentially with . The "exponential" here means that each previous input value contributes times smaller to the output than .

This exponential character of the decline of weights means that, if to try to implement an equivalent filter as a nonrecursive filter, then an infinite number of preceding input values should be taken into account (and this is, strictly speaking, computationally impossible). This feature illustrates a major advantage of recursive filters over nonrecursive filters - computational simplicity.