Algebraic formulation of the fast Fourier transform

A new unified formulation of the fast Fourier transform is presented. It is shown that all FFT algorithms can be derived by different methods of multidimensional array unwrapping. The eight principal decimation in time FFT algorithms are derived. Two algorithms with the desirable properties of sequential input, sequential output and identical computational geometry are also derived. The derivation of the decimation in frequency FFT algorithms is then discussed. While most of the results presented here have been derived earlier using matrix Kronecker products, the present formulation is simpler and more intuitive that the equivalent matrix formulation.