BIBO stability in the presence of nonessential singularities of the second kind in 2-D digital filters

In this paper, the open problem regarding the BIBO stability of two-dimensional linear shift invariant filters, in the presence of nonessential singularities of the second kind, is considered. Necessary and sufficient conditions for boundedness, and
l_2
and
l_1
stabilities of a function
G(z_1, z_2)= P(z_1, z_2)/[Q(z_1, z_2)]^n
, where
P/Q
has simple nonessential singularities of the second kind on
T^2
, are obtained. These conditions are expressed in a very simple way in terms of the multiplicity of the zeros of certain resultants of two-variable polynomials. Many illustrative examples are also given.