International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. B, ch. 1.3, pp. 7273

A finite set of reflections can be periodized without aliasing by the translations of a suitable sublattice of the reciprocal lattice ; the converse operation in real space is the sampling of ρ at points X of a grid of the form (Section 1.3.2.7.3). In standard coordinates, is periodized by , and is sampled at points .
In the absence of symmetry, the unique data are
They are connected by the ordinary DFT relations: or and or
In the presence of symmetry, the unique data are
– or in real space (by abuse of notation, D will denote an asymmetric unit for x or for m indifferently);
– in reciprocal space.
The previous summations may then be subjected to orbital decomposition, to yield the following `crystallographic DFT' (CDFT) defining relations: with the obvious alternatives in terms of . Our problem is to evaluate the CDFT for a given space group as efficiently as possible, in spite of the fact that the group action has spoilt the simple tensorproduct structure of the ordinary threedimensional DFT (Section 1.3.3.3.1).
Two procedures are available to carry out the 3D summations involved as a succession of smaller summations:
Clearly, a symmetry expansion to the largest fully reducible subgroup of the space group will give maximal decomposability, but will require computing more than the unique results from more than the unique data. Economy will follow from factoring the transforms in the subspaces within which the space group acts irreducibly.
For irreducible subspaces of dimension 1, the group action is readily incorporated into the factorization of the transform, as first shown by Ten Eyck (1973).
For irreducible subspaces of dimension 2 or 3, the ease of incorporation of symmetry into the factorization depends on the type of factorization method used. The multidimensional Cooley–Tukey method (Section 1.3.3.3.1) is rather complicated; the multidimensional Good method (Section 1.3.3.3.2.2) is somewhat simpler; and the Rader/Winograd factorization admits a generalization, based on the arithmetic of certain rings of algebraic integers, which accommodates 2D crystallographic symmetries in a most powerful and pleasing fashion.
At each stage of the calculation, it is necessary to keep track of the definition of the asymmetric unit and of the symmetry properties of the numbers being manipulated. This requirement applies not only to the initial data and to the final results, where these are familiar; but also to all the intermediate quantities produced by partial transforms (on subsets of factors, or subsets of dimensions, or both), where they are less familiar. Here, the general formalism of transposition (or `orbit exchange') described in Section 1.3.4.2.2.2 plays a central role.
References
Ten Eyck, L. F. (1973). Crystallographic fast Fourier transforms. Acta Cryst. A29, 183–191.Google Scholar