A set of necessary stability conditions for \(m\)-D nonlinear digital filters (Q1898702)

The intrinsic feature of the so-called \(m\)-D systems is that these propagate information in \(m\) separate directions, for example, \(i_1, i_2, \dots, i_m\). Here asymptotic stability of possibly nonlinear, recursive, first hyperquadrant causal (non-singular) \(m\)-D digital systems (filters) is considered. Instead of a given \(m\)-D system, its projections on all possible direction subsets \(\{j_1, j_2, \dots, j_\upsilon\} \subset \{i_1, i_2, \dots, i_m \}\), \(1\leq \nu< m\), termed \(\nu\)-D subsystems (projections), are examined. This is clearly a considerable simplification of a system structure so that only necessary conditions may be provided. In effect, a total of \(2^m -2\) necessary conditions is developed, \(m\) of which are of 1-D nature. The author refers also his results to the well known notion of the so-called practical stability introduced by Agathoklis and Burton.