On the invariance of second-order modes under frequency transformations in 2-D separable denominator digital filters

摘要

This paper discusses the invariance of second-order modes under frequency transformations in 2-D separable denominator digital filters. In 1-D digital filters, it is known that the second-order modes which characterize minimum roundoff noise are invariant under frequency transformations. This paper proves that the second-order modes of 2-D separable denominator digital filters are completely invariant under frequency transformations if and only if the special conditions are satisfied. When the conditions are not satisfied, the horizontal second-order modes are invariant under horizontal frequency transformations, and the vertical second-order modes are invariant under vertical frequency transformations. Furthermore, when lowpass-to-bandpass or lowpass-to-bandstop transformations are applied under the fixed bandwidth, the second-order modes are completely invariant even in 2-D separable denominator digital filters.