Sufficient conditions for the nonexistence of limit cycles in two-dimensional digital filters

Abstract

The stability of the two classes of two-dimensional digital filters defined by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F1: x_{i+1,j+1} = Q_R[ax_{i+1,j}+bx_{i,j+1}+cx_{i,j}]</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F2: x_{i+1,j+1} = Q_R[ax_{i+1,j}]+Q_R[bx_{i,j+1}]</tex> is studied. Here <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q_R</tex> is the rounding operator, and fixed-point arithmetic is used. Sufficient conditions for the stability of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F1</tex> and necessary and sufficient conditions for the stability of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F2</tex> are derived. For the more general case of higher order two-dimensional (2-D) digital filters, sufficient conditions for the nonexistence of separable 2-D limit cycles are derived by extending the results of Claasen et al. [1].