Multidimensional DFT processing in subspaces whose dimensions are relatively prime

Practical algorithms for computing the discrete Fourier transform (DFT) result from the Kronecker product of small N algorithms. This paper shows that several options exist for indexing of the input and output of this DFT. If the Chinese remainder theorem (CRT) is used to expand the output index, then an alternate integer representation (AIR) is shown to determine the input index. It is shown that the roles of the CRT and AIR can be reversed so that the CRT and AIR determine input and output indices, respectively. As a consequence of the indexing the DFT must be processed in subspaces whose dimensions are relatively prime.