Fast algorithms for signal processing using finite field operations

A signal processing system employing digital arithmetic can be made equivalent to a set of interconnected subsystems each using finite field operations to alleviate effects of round-off noise, particularly in recursive structures. The realization of the subsystems involves the multiplication of polynomials over a finite field. Cyclic convolutions may be represented as polynomial multiplications, in a residue class ring modulo the polynomial (x
^k
-1). Fast algorithms arise when the ring is decomposed into a direct product using its minimal ideals, the orthogonality relationships between the idempotents of the minimal ideals resembling the decomposition properties of the Chinese remainder theorem for real polynomials. The resulting component polynomial multiplications involve elements with smaller degrees. A transform domain, based upon an n
^th
root of unity in an extension field, offers an alternative domain for performing the component operations. Polynomial multiplications are equivalent to the vector inner product of the corresponding transform coefficients. The minimal ideal components, when mapped into the transform domain, have nonzero coefficients only in predetermined subsets, the cyclotomic subsets, which permits one coefficient to generate all others by successively forming its powers.