Efficient recursive realizations of FIR filters

Abstract

Recursive filter structures have been found for FIR filters with piecewisepolynomial or piecewise-(polynomial · sinusoid) impulse responses. The amount of arithmetic required for these filters is proportional to the number of piecewise sections in their impulse responses rather than the actual filter lengths. In this paper, it is shown that these impulse response expressions are quite good approximations to many practical filters. Low-pass filters, high-pass filters, narrowband, band-pass, and band-stop filters, Hilbert transformers, and differentiators all have impulse responses which can be approximated by these forms, and a long filter impulse response consists of only a few piecewise sections with greatly reduced arithmetic requirements. Though this technique is based on time-domain approximation, a frequency-domain optimization to select filter parameters is presented with excellent results.