Recursive Filter:
In recursive filters , the output at the moment is a function of the output values at the previous moments and, probably, of the input values.
A major advantage of recursive filters over nonrecursive filters is that they are computationally simpler. For example, for any recursive linear filter , there is a functionally equivalent nonrecursive linear filter. But the nonrecursive variant may be characterized by an infinitely large window size and, hence, be difficult for practical implementation. (See, for example, explanations on the exponential filter ).
Mathematically, a recursive filter may be described by the following general expression:
where

is an arbitrary function of the previous output values and input values , linear or nonlinear.
Some examples of recursive filters are the exponential filter , the Kalman filter .