Covariance-invariant digital filtering

Abstract

When discretizing continuous-time filters, one is often interested in preserving a property termed covariance-invariance. Techniques are outlined for synthesizing discrete-time filters which are covariance-invariant with corresponding continuous-time filters. The synthesis techniques involve straightforward matrix decompositions or polynomial root-finding algorithms that can easily be programmed on a digital computer. Applications of the technique to digital filter synthesis are outlined, with example designs presented for covariance-invariant Butterworth and Chebyshev digital filters. Based on the frequency response of these designs it is argued that the method of covariance-invariance is superior to the methods of impulse-invariance and bilinear-z as a response matching design technique for the synthesis of digital filters. This superiority is especially apparent at sampling rates that are marginal with respect to filter critical frequencies. Moreover, the covariance-invariant designs are stably invertible solutions to a so-called spectral factorization problem. This property may be important in inverse filtering applications.