Derivation of Recursive Digital Filters by the Step-Invariant and the Ramp-Invariant Transformations

摘要

This document describes two procedures for designing recursive digital filters from continuous-time filters when the ratio of the sampling frequency to the pole frequency is small. The coefficients of the proposed digital filters, which are derived from the step and ramp invariance of the corresponding analog filters, have been determined for real and complex poles. For higher-order filters realized in a parallel form, it is demonstrated that the discrete-time transfer function of digital filters obtained by the step and ramp invariance can be derived directly from the standard z-transformation or from the partial fraction expansion of the continuous-time transfer function. The discrete-time transfer functions of the step- and ramp-invariant filters realized in a cascade form have also been derived. Finally, the performance of these methods is demonstrated by plotting the magnitude ahd phase responses of the first- and second-order digital filters. For high-order Butterworth and elliptic filters, the magnitude responses of step-invariant and ramp-invariant filters are compared with those obtained by usual methods such as the standard z and the bilinear transformations.