Properties of the multidimensional generalized discrete Fourier transform

In this work the generalized discrete Fouriertransform (GFT), which includes the DFT as a particular case, is considered. Two pairs of fast algorithms for evaluating amultidimensional GFT are given (T-algorithm, F-algorithm, and T'-algorithm, F'-algorithm) It is shown that in the case of the DFT of a vector, the T-algorithm represents a form of the classical FFT algorithm based on a decimation in time, and the F-algorithm represents a form of the classical FFT algorithm based ondecimation in frequency. Moreover, it is shown that the T'-algorithm and the T-algorithm involve exactly the same arithmetic operations on the same data. The same property holds for the F'-algorithm and the F-algorithm. The relevance of such such algorithms is discussed, and it is shown that the T'-algorithm and the F'-algorithm are particularly advantageous for evaluating the DFT of large sets of data.