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 non-recursive linear filter. But the non-recursive 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 .