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 nonrecursivefilters 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.