Synthesis of 2-D separable denominator digital filters with minimum roundoff noise and no overflow oscillations

Abstract

This paper presents a new expression of the variance of roundoff noise in 2-D separable denominator digital filters described by Roesser's local state-space model. The covariance matrices and noise matrices necessary to analyze the variance of roundoff noise under the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1_2</tex> norm scaling are obtained as the solutions of Liapunov equations. The synthesis problem of 2-D separable denominator digital filters with minimum roundoff noise are formulated under the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1_2</tex> norm scaling. The synthesis method of 2-D minimum noise realizations under the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1_2</tex> norm scaling is proposed by applying the minimization technique of roundoff noise in the 1-D case. Furthermore, it is shown that 2-D minimum noise realizations are rotated and scaled balanced realizations. Using this property, 2-D minimum noise realizations are proved to be free of overflow oscillations under zero input conditions, if their second-order modes are distinct.