Invariance of Second-Order Modes under Frequency Transformation in 2-D Separable Denominator Digital Filters

This paper discusses the invariance of the second-order modes of 2-D separable denominator digital filters under frequency transformation. This paper first derives a state-space description for 2-D digital filters obtained by 2-D frequency transformation and then represents the controllability Gramians and the observability Gramians of the transformed 2-D digital filters. This description proves that the second-order modes of 2-D separable denominator digital filters are not invariant under all frequency transformations, but invariant under specific frequency transformations of which transfer functions are strictly proper. These frequency transformations include lowpass-bandpass and lowpass-bandstop frequency transformations keeping the same bandwidth as that of the prototype lowpass filter. It is also shown that the horizontal second-order modes are invariant when only horizontal transformation is applied, and the vertical second-order modes are invariant when only vertical transformation is applied. This paper further remarks significance brought by the invariance of the second-order modes under frequency transformation.