Decomposition of High-Order FIR Filters and Minimum-Phase Filter Design

In this study, the implementation of high-order FIR filter decomposition and minimum-phase filter design is investigated. One method is presented for decomposing arbitrary linear-phase FIR filters with distinct roots into the cascade of first-order, second-order and fourth-order subfilters. The other method is described for transforming nonrecursive filters with even-order, equal-ripple attenuation in the pass-band, stop-band and linear-phase into those with minimum-phase and half the degree, and again with equal-ripple attenuation in the pass-band and stop-band. The technique consists of quick and accurate polynomial root finding of the z -domain filter transfer function by searching a finite region in the complex z-plane, and separating the zeros in the complex z -domain. In FIR filter decomposition, the search of roots to determine the subfilter impulse response coefficients is restricted to distinct roots in four regions in the complex z -plane: on the real axis, on the unit circle, inside the unit circle and at (1, 0) or (-1, 0). In minimum-phase filter design, the search of roots is restricted in two categories: on the unit circle and inside the unit circle. In both methods, we used Lang’s root finding program to get the zeros of the FIR filter.

Arbitrary FIR filters were designed and decomposed for all possible orders of subfilters. FIR filters with even-order, zero-phase and equal-ripple were designed and generated the half degree minimum-phase filters. Both methods have been tested on FIR filters with orders ranging to over 500 and have proven effective in decomposing filters to the cascade realization and designing minimum-phase filters.