Round-Off Noise Free Recursive Finite Impulse Filters on a Finite Ring

Hideo MURAKAMI  Irving S. REED  

IEICE TRANSACTIONS (1976-1990)   Vol.E60   No.9   pp.451-458
Publication Date: 1977/09/25
Online ISSN: 
Print ISSN: 0000-0000
Type of Manuscript: PAPER
Category: Circuit and Systems

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Fourier-like transforms over the Galois field, GF(q2), and the direct sum of such Galois fields, analogous to the field of complex numbers, were developed and used to design round-off noise free nonrecursive filters (1), (2). It was shown (3), (4) that recursive realizations of finite impulse response (FIR) filters on the finite field or ring are also possible by techniques, analogous to those recursive realizations used on the usual complex number field. This paper developes an algorithm to reduce the number of recursive equations needed to realize a filter when recursive equations are restricted to be first order. The recursive filters developed here do not have the accumulation of round-off or truncation error one usually expects in recursive computations. The FIR filter without error is more naturally a digital" filter than the usual FIR digital filter. By the very finiteness of the finite field or ring, only finite bit words are needed to realize the FIR filter. Such digital filters when realized as discrete transducers are information lossless in the sense of Shannon (5, pp. 26).