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

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 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 norm scaling. The synthesis method of 2-D minimum noise realizations under the 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.