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Showing papers on "Split-radix FFT algorithm published in 1970"


Journal ArticleDOI
TL;DR: This paper derives explicit expressions for the mean square error in the FFT when floating-point arithmetics are used, and upper and lower bounds for the total relative meansquare error are given.
Abstract: The fast Fourier transform (FFT) is an algorithm to compute the discrete Fourier coefficients with a substantial time saving over conventional methods. The finite word length used in the computer causes an error in computing the Fourier coefficients. This paper derives explicit expressions for the mean square error in the FFT when floating-point arithmetics are used. Upper and lower bounds for the total relative mean square error are given. The theoretical results are in good agreement with the actual error observed by taking the FFT of data sequences.

89 citations


Journal ArticleDOI
TL;DR: It is concluded that the Blackman-Tukey technique is more effective than the FFT approach in computing power spectra of short historic time series, but for long records the fast Fourier transform is the only feasible approach.
Abstract: Since controversy has arisen as to whether the Blackman-Tukey or the fast Fourier transform (FFT) technique should be used to compute power spectra, single and cross spectra have been computed by each approach for artificial data and real data to provide an empirical means for determining which technique should be used. The spectra were computed for five time series, two sets of which were actual field data. The results show that in general the two approaches give similar estimates. For a spectrum with a large slope, the FFT approach allowed more window leakage than the Blackman-Tukey approach. On the other hand, the Blackman-Tukey approach demonstrated a better window closing capability. From these empirical results it is concluded that the Blackman-Tukey technique is more effective than the FFT approach in computing power spectra of short historic time series, but for long records the fast Fourier transform is the only feasible approach.

19 citations


Journal ArticleDOI
TL;DR: New and simple derivations for the two basic FFT algorithms are presented that provide an intuitive basis for the manipulations involved and reduce the operation to the calculation of a large number of simple two-data-point transforms.
Abstract: The fast Fourier transform (FFT) provides an effective tool for the calculation of Fourier transforms involving a large number of data points. The paper presents new and simple derivations for the two basic FFT algorithms that provide an intuitive basis for the manipulations involved. The derivation for the "decimation in time" algorithm begins with a crude analysis for the zero frequency and fundamental components using only two data samples, one at the beginning and the second at the midpoint of the period of interest. Successive interpolations of data points midway between those previously used result in a refinement of the amplitudes already determined and a first value for the next higher order coefficients. The derivation of the "decimation in frequency" algorithm begins by resolving the original data set into two new data sets, one whose transform includes only even harmonic terms and a second whose transform includes only odd harmonic terms. Since the first of the two new data sets repeats after the midpoint, it can be transformed using only the first half of the data points. The second of the new data sets is multiplied by the negative fundamental function, thereby reducing its order by one and converting it into a data set that transforms into even harmonics only; in this form it can also be transformed using only the first half of the data set. Successive applications of this procedure result finally in reducing the operation to the calculation of a large number of simple two-data-point transforms.

12 citations


Journal ArticleDOI
TL;DR: A simple fast Fourier transformation (FFT) algorithm has been specifically adapted to calculate the experimental radial distribution function and its greatest advantage is its internal consistency—the ability to exactly transform back to the original domain.
Abstract: A simple fast Fourier transformation (FFT) algorithm has been specifically adapted to calculate the experimental radial distribution function. The number of equi-spaced data points must be a power of two [N = 2n for integer n] and must be greater than the Nyquist frequency [N = 2(rmax) (smax)/2π]. When properly defined, the data set is expanded as an odd function. The greatest advantage of the FFT algorithm is its internal consistency—the ability to exactly transform back to the original domain.

6 citations