Convergence behavior and convergence rates of stack filters

The convergence behavior of type-0 through type-3 stack filters is investigated. It is shown that stack filters of type-0 through type-2 all possess the convergence property; that is, they filter any input signal to a root after consecutive passes of the filter under any appending strategy. A counterexample is given to show that not all type-3 stack filters have this convergence property. The rate of convergence for convergent stack filters is also shown. It is shown that stack filters of type-0 will take at most a single pass to filter any input signal to a root. The rate of convergence of type-1 and type-2 stack filters is shown to be linear in the length of the input signal