Multidimensional Index Mapping *

A change of index variable or an index mapping is used to uncouple the calculations of the discrete Fourier transform (DFT). This can result is a signi cant reduction in the required arithmetic and the resulting algorithm is called the fast Fourier transform (FFT). A powerful approach to the development of e cient algorithms is to break a large problem into multiple small ones. One method for doing this with both the DFT and convolution uses a linear change of index variables to map the original one-dimensional problem into a multi-dimensional problem. This approach provides a uni ed derivation of the Cooley-Tukey FFT, the prime factor algorithm (PFA) FFT, and the Winograd Fourier transform algorithm (WFTA) FFT. It can also be applied directly to convolution to break it down into multiple short convolutions that can be executed faster than a direct implementation. It is often easy to translate an algorithm using index mapping into an e cient program. The basic de nition of the discrete Fourier transform (DFT) is