Survey of new immittance-type stability tests for two-dimensional digital filters

The paper brings a brief report on three new algebraic tests to determine whether a two variable polynomial has all its zeros inside the unit bi-circle (is 'stable'). A three-term recursion algorithm associates the tested two-variable polynomial with a sequence of matrices (the 2-D 'table') that possess a symmetry which allows to compute only half of the entries for each matrix. The recursions incorporate a deconvolution/division mechanism that removes recursively redundant common polynomial factors and prevents an exponential grow of the raw dimension of the matrices. A minimal set of conditions necessary and sufficient for stability for a polynomial with variables of degrees (n₁, n₂) requires one or two 1-D tests of degree n₁ and the testing of whether a last 1-D polynomial has zeros in the real interval [-1,1] or on |z| = 1. The degree of this last polynomial is only 2n₁n₂.