TABLE OF CONTENTS In this section, we will present different stability criteria for causal and semi-causal recursive filters. A stability criterion is a sufficient and necessary condition to assure BIBO stability.
We start by introducing three subsets of 
:
The open unit bidisk 
The closed unit bidisk 
The unit bi-circle or toric unit 
Here are the three subsets of 
:
The open unit disk 
The closed unit disk 
The unit circle 
Let us consider a continuous function 
that does not vanish in the closed unit bidisk. It results from the continuity of f and the fact that 
is a closed and bounded domain, that the minimal value ? taken by f(z) on 
is attained by (at least) an element 
and since f does not vanish in , we have ? > 0.
According to its continuity property, the function f cannot increase suddenly from ?>0 when z leaves the closed unit bidisk. From there, we can extend to an open bidisk containing the set of points where f does not vanish: there exist ? 1>0 and ? 2>0, so that:
Let us consider a causal recursive filter of transfer function H(z) on the convergence domain C. We write B(z)/A(z) [7] the irreducible rational fraction coinciding with H(z) on C. Since the filter is causal, that means that 
.
THEOREM 11.1. (the Rudin and Shanks Theorem) let a...
