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Showing papers on "Prime-factor FFT algorithm published in 1968"


Journal ArticleDOI
01 Jun 1968
TL;DR: The discrete Fourier transform of a sequence of N points, where N is a prime number, is shown to be essentially a circular correlation, which permits the discrete Fouriers transform to be computed by means of a fast Fouriertransform algorithm, with the associated increase in speed, even though N is prime.
Abstract: The discrete Fourier transform of a sequence of N points, where N is a prime number, is shown to be essentially a circular correlation. This can be recognized by rearranging the members of the sequence and the transform according to a rule involving a primitive root of N. This observation permits the discrete Fourier transform to be computed by means of a fast Fourier transform algorithm, with the associated increase in speed, even though N is prime.

523 citations


Journal ArticleDOI
Glenn D. Bergland1
TL;DR: In this article, a new procedure for calculating the complex, discrete Fourier transform of real-valued time series is presented for an example where the number of points in the series is an integral power of two.
Abstract: A new procedure is presented for calculating the complex, discrete Fourier transform of real-valued time series. This procedure is described for an example where the number of points in the series is an integral power of two. This algorithm preserves the order and symmetry of the Cooley-Tukey fast Fourier transform algorithm while effecting the two-to-one reduction in computation and storage which can be achieved when the series is real. Also discussed are hardware and software implementations of the algorithm which perform only (N/4) log2 (N/2) complex multiply and add operations, and which require only N real storage locations in analyzing each N-point record.

134 citations


Journal ArticleDOI
TL;DR: The base 8 algorithms described in this paper allow one to perform as many base 8 iterations as possible and then finish the computation by performing a base 4 or a base 2 iteration if one is required, which preserves the versatility of the base 2 algorithm while attaining the computational advantage of thebase 8 algorithm.
Abstract: 1. Introduction. Cooley and Tukey stated in their original paper [1] that the Fast Fourier Transform algorithm is formally most efficient when the number of samples in a record can be expressed as a power of 3 (i.e., N = 3m), and further that there is little efficiency lost by using N = 2m or N = 4™. Later, however, it was recognized that the symmetries of the sine and cosine weighting functions made the base 4 algorithms more efficient than either the base 2 or the base 3 algorithms [2], [3]. Making use of this observation, Gentleman and Sande have constructed an algorithm which performs as many iterations of the transform as possible in a base 4 mode, and then, if required, performs the last iteration in a base 2 mode. Although this "4 + 2" algorithm is more efficient than base 2 algorithms, it is now apparent that the techniques used by Gentleman and Sande can be profitably carried one step further to an even more efficient, base 8 algorithm. The base 8 algorithms described in this paper allow one to perform as many base 8 iterations as possible and then finish the computation by performing a base 4 or a base 2 iteration if one is required. This combination preserves the versatility of the base 2 algorithm while attaining the computational advantage of the base 8 algorithm.

88 citations