Properties of the Multidimensional Generalized Discrete Fourier Transform

In this work the generalized discrete Fourier transform (GFT), which includes the DFT as a particular case, is considered. Two pairs of fast algorithms for evaluating a multidimensional 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 on decimation 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 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.