Vector computation of the discrete Fourier transform
Since the invention of fast algorithms for the computation of the discrete Fourier transform (DFT) by Cooley and Tukey (1965), the DFT has been widely used for the frequency-domain analysis and design of signals and systems in communications, digital signal processing, and in many other areas of science and engineering. While the Cooley-Tukey algorithms are simple, regular, and efficient, they have the drawback of requiring a significant amount of overhead operations such as bit-reversal, data-swapping, etc. In this paper, we develop a generalized version of a new family of DFT algorithms by decomposing a form of the DFT relation in which the input data and transform quantities are represented as vectors. These algorithms have the features that eliminate or reduce the drawbacks of the Cooley-Tukey algorithms while improving the simplicity and regularity of their implementations. The generalized version makes it easier to deduce a large family of algorithms with different features. The relative merits of the various algorithms with different vector lengths are discussed and the optimum vector length for DFT computation is pointed out.