Invariance of second-order modes of 2-D digital filters under 2-D frequency transformation

The so-called second-order modes, which are the square roots of the eigenvalues of the product of the controllability Gramian and observability Gramian of the filters, scale the minimal attainable value of roundoff noise of one-dimensional (1-D) and two-dimensional (2-D) filters. This paper carries out a preliminary investigation on the invariance property of the second-order modes of 2-D digital filters under 2-D frequency transformation based on the state-space formulation of 2-D frequency transformation proposed by the co-authors. Some nontrivial numerical examples are given to show that, compared with the results given in the 1-D frequency transformation for 2-D separable digital filters case, the similar invariance property may also hold for the 2-D frequency transformation of 2-D digital filters, i.e., the horizontal second-order modes keep invariant when only horizontal 2-D transformation is applied while the vertical second-order modes keep invariant when only vertical 2-D transformation is applied.