Design of ARMA digital filters by pole-zero decomposition

Abstract

A new technique for design of digital filters is presented in this paper. The technique exploits the spectral approximation property of autoregressive modeling to reduce ripple at the edges of the transition band in the filter response. An autoregressive model approximates a given spectrum better at the peaks than at the Valleys. Spectral information around the transition band is transformed into peaks by splitting the given squared-magnitude frequency response into two component spectra. This splitting is accomplished using a pole-zero decomposition technique, which in turn uses the properties of the derivative of phase spectrum of minimum phase filters. One of the component spectra corresponds nearly to the response of an all-pole filter, and the other component spectrum corresponds nearly to the response of an all-zero filter. Each of these corresponends spectra can be represented by a small number of coefficients using autoregressive modeling. The resulting two sets of autoregressive coefficients determine poles and zeros of the autoregressive moving average (ARMA) digital filter. Ripple characteristics in the response of the ARMA filter can be controlled by appropriately choosing the number of poles and zeros. It is shown that a wide variety of magnitude frequency response characteristics can be approximated by an ARMA filter of low order using this technique. The technique does not work well in cases where spectral approximation by autoregressive modeling is poor. Such cases arise when the component spectra have very large dynamic range.