A recursive variant of Mallows' filter class

Mallows' (1980) nonlinear FIR filter class includes most of the nonlinear digital filters in wide use today. This filter class is defined in terms of five axioms, and this paper modifies two of these axioms to obtain a class of symmetric recursive nonlinear filters. One of the main points considered in this paper is the consequence of Mallows' location invariance axiom A2, and general prescriptions are given for both Mallows' original FIR filter class and its recursive extension. This representation is also extremely useful in discussing the root sequences of both the recursive and nonrecursive structures. Since recursive filters can exhibit various forms of instability, a number of stability results are presented for this class of filters. Finally, some simple examples are considered that illustrate the potential utility of the results presented.