Convergence Behavior and Root Signal Sets of Stack Filters*

The four different types of stack filters, type-0 through type-3, are determined by four different shapes of the on-set of the positive Boolean function from which the stack filter is constructed. Under all three appending strategies commonly considered in the literature---first and last value carry-on strategy, constant value carry-on strategy, and the circular approach--stack filters of type-0 through type-2 possess the convergence property, while type-3 stack filters do not all share this property. Examples of cyclic behavior in type-3 stack filters are given. Conditions under which certain operations on stack filters which possess the convergence property produce other filters with this property are provided. In perhaps the most important result in this paper, it is shown that the root signal set of any type-3 stack filter is the intersection of the root sets of the type-1 and type-2 stack filters from which the type-3 filter is constructed. This should simplify the task of finding the set of roots of type-3 stack filters. The rates of convergence for stack filters of type-1 and type-2 are determined for each appending approach. The convergence behavior and rates of convergence of stack filters of type-1 and type-2 are then generalized to include type-1 and type-2 filters with index i.