Causal invertibility and stability of asymmetric half-plane digital filters

In this paper the following question is studied: suppose
R \in l_{1}(Z^{2})
, and let S
h
denote the semigroup generated by the support of h; when does h have an inverse in l
1
(S
h
)? From an engineering viewpoint the question can be asked as follows: When does a causal stable filter have a causal stable inverse? In this paper we develop a mathematical framework for studying this question and derive a general necessary and sufficient condition for an affirmative answer. It is shown that this condition unifies several previously known criteria for the stability of asymmetric half-plane filters and also gives somewhat sharper conclusions. Extensions to n-dimensional filters and to feedback stability are also presented.