International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 5657

The cyclic convolutions generated by Rader's multiplicative reindexing may be evaluated more economically than through DFTs if they are reexamined within a new algebraic setting, namely the theory of congruence classes of polynomials [see, for instance, Blahut (1985), Chapter 2; Schroeder (1986), Chapter 24].
The set, denoted , of polynomials in one variable with coefficients in a given field has many of the formal properties of the set of rational integers: it is a ring with no zero divisors and has a Euclidean algorithm on which a theory of divisibility can be built.
Given a polynomial , then for every there exist unique polynomials and such thatand is called the residue of modulo . Two polynomials and having the same residue modulo are said to be congruent modulo , which is denoted by
If is said to be divisible by . If only has divisors of degree zero in , it is said to be irreducible over (this notion depends on ). Irreducible polynomials play in a role analogous to that of prime numbers in , and any polynomial over has an essentially unique factorization as a product of irreducible polynomials.
There exists a Chinese remainder theorem (CRT) for polynomials. Let be factored into a product of pairwise coprime polynomials [i.e. and have no common factor for ]. Then the system of congruence equationshas a unique solution modulo . This solution may be constructed by a procedure similar to that used for integers. LetThen and are coprime, and the Euclidean algorithm may be used to obtain polynomials and such thatWith , the polynomialis easily shown to be the desired solution.
As with integers, it can be shown that the 1:1 correspondence between and sends sums to sums and products to products, i.e. establishes a ring isomorphism:
These results will now be applied to the efficient calculation of cyclic convolutions. Let and be two vectors of length N, and let be obtained by cyclic convolution of U and V:The very simple but crucial result is that this cyclic convolution may be carried out by polynomial multiplication modulo : ifthen the above relation is equivalent toNow the polynomial can be factored over the field of rational numbers into irreducible factors called cyclotomic polynomials: if d is the number of divisors of N, including 1 and N, thenwhere the cyclotomics are well known (Nussbaumer, 1981; Schroeder, 1986, Chapter 22). We may now invoke the CRT, and exploit the ring isomorphism it establishes to simplify the calculation of from and as follows:
When N is not too large, i.e. for `short cyclic convolutions', the are very simple, with coefficients 0 or ±1, so that (i) only involves a small number of additions. Furthermore, special techniques have been developed to multiply general polynomials modulo cyclotomic polynomials, thus helping keep the number of multiplications in (ii) and (iii) to a minimum. As a result, cyclic convolutions can be calculated rapidly when N is sufficiently composite.
It will be recalled that Rader's multiplicative indexing often gives rise to cyclic convolutions of length for p an odd prime. Since is highly composite for all other than 23 and 47, these cyclic convolutions can be performed more efficiently by the above procedure than by DFT.
These combined algorithms are due to Winograd (1977, 1978, 1980), and are known collectively as `Winograd small FFT algorithms'. Winograd also showed that they can be thought of as bringing the DFT matrix F to the following `normal form':where
The elements on the diagonal of B can be shown to be either real or pure imaginary, by the same argument as in Section 1.3.3.2.3.1. Matrices A and C may be rectangular rather than square, so that intermediate results may require extra storage space.
References
Blahut, R. E. (1985). Fast Algorithms for Digital Signal Processing. Reading: AddisonWesley.Nussbaumer, H. J. (1981). Fast Fourier Transform and Convolution Algorithms. Berlin: SpringerVerlag.
Schroeder, M. R. (1986). Number Theory in Science and Communication, 2nd ed. Berlin: SpringerVerlag.
Winograd, S. (1977). Some bilinear forms whose multiplicative complexity depends on the field of constants. Math. Syst. Theor. 10, 169–180.
Winograd, S. (1978). On computing the discrete Fourier transform. Math. Comput. 32, 175–199.
Winograd, S. (1980). Arithmetic Complexity of Computations. CBMSNST Regional Conf. Series Appl. Math, Publ. No. 33. Philadelphia: SIAM Publications.