It is well known that the magnitudes of the coefficients of the discrete
Fourier transform (DFT) are invariant under certain operations on the
input data. In this paper, the effects of rearranging the elements of an
input data on its DFT are studied. In the one-dimensional case, the
effects of permuting the elements of a finite sequence of length N on
its discrete Fourier transform (DFT) coefficients are investigated. The
permutations that leave the unordered collection of Fourier coefficients
and their magnitudes invariant are completely characterized. Conditions
under which two different permutations give the same DFT coefficient
magnitudes are given. The characterizations are based on the
automorphism group of the additive group Z_{N}
of integers modulo *N* and the group of translations of Z_{N}.
As an application of the results presented, a generalization of the
theorem characterizing all permutations that commute with the discrete
Fourier transform is given. Numerical examples illustrate the obtained
results. Possible generalizations and open problems are discussed. In
higher dimensions, results on the effects of certain geometric
transformations of an input data array on its DFT are given and
illustrated with an example.