Polynomial Residue Systems via Unitary Transforms

A polynomial, A(z), can be represented by a polynomial residue system and, given enough independent residues, the polynomial can be reconstituted from its residues by the Chinese remainder theorem (CRT). A special case occurs when the discrete Fourier transform and its inverse realise the residue evaluations and CRT respectively, in which case the residue system is realised by the action of a matrix transform that is unitary. In this paper we generalise the class of residue systems that can be realised by the action of unitary transforms beyond the Fourier case, by suitable modification of the polynomial, A(z). We identify two new types of such system that are of particular interest, and also extend from the univariate to the multivariate case. By way of example, we show how the generalisation leads to two new types of complementary array pair. 1. Polynomial Residue Systems Let A(z) = (A0 + A1z + . . . + AN−1z) be a univariate polynomial with coefficients A = (A0, A1, . . . , AN−1) ∈ C , for C the field of complex numbers. One can embed A(z) in a polynomial modulus M(z), where A(z) = A(z) mod M(z), iff deg(M(z)) ≥ N, where deg(∗) is the algebraic degree of ∗. Let M(z) = ∏m−1 j=0 mj(z) be the product of m mutually-prime polynomials. Then m residues can be extracted from A(z), A(z)⇔ (A(z) mod m0(z), A(z) mod m1(z), . . . , A(z) mod mm−1(z)). (1) The conversion from left to right in (1) is the evaluation of the residues of A(z) with respect to the residue system described by the factors of M(z). If deg(M) ≥ N , and on condition that the mj(z) are mutually prime, this conversion is invertible, and then the conversion from right to left in (1) is the reconstruction of A(z) from its residues by the Chinese remainder theorem (CRT).