Gramian-Preserving Frequency Transformation and Its Application to Analog Filter Design

This paper presents a new state-space formulation of the frequency transformation of linear continuous-time systems. We call this formulation “Gramian-preserving frequency transformation,” which enables us to convert a given prototype state-space system into other systems that have the same controllability/observability Gramians as those of the prototype system. As a practical application of this result, we propose a new method for the design of analog filters of which basic building blocks are integrators. It is demonstrated that the analog filters designed by the Gramian-preserving frequency transformation have the same structural properties with respect to the Gramians as those of the prototype filter, and that the performance such as the dynamic range and the sensitivity of the designed filters is retained even if the filter specification is changed. In addition to presenting this result, we also address the physical meaning of the Gramian-preserving frequency transformation. It is shown that the Gramian-preserving frequency transformation is derived from the state-space model that is obtained by replacing each integrator of a prototype filter with a reactance function, where the structure of this reactance function is given as the scaled version of the Foster canonical form.