FFTW is an implementation of the discrete Fourier transform (DFT) that adapts to the hardware in order to maximize performance. This paper shows that such an approach can yield an implementation that is competitive with hand-optimized libraries, and describes the software structure that makes our current FFTW3 version flexible and adaptive. We further discuss a new algorithm for real-data DFTs of prime size, a new way of implementing DFTs by means of machine-specific single-instruction, multiple-data (SIMD) instructions, and how a special-purpose compiler can derive optimized implementations of the discrete cosine and sine transforms automatically from a DFT algorithm.

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The Design and Implementation of FFTW3

FFT literature has been mostly concerned with minimizing the number of floating-point operations performed by an algorithm. Unfortunately, on present-day microprocessors this measure is far less important than it used to be, and interactions with the processor pipeline and the memory hierarchy have a larger impact on performance. Consequently, one must know the details of a computer architecture in order to design a fast algorithm. In this paper, we propose an adaptive FFT program that tunes the computation automatically for any particular hardware. We compared our program, called FFTW, with over 40 implementations of the FFT on 7 machines. Our tests show that FFTW's self-optimizing approach usually yields significantly better performance than all other publicly available software. FFTW also compares favorably with machine-specific, vendor-optimized libraries.

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FFTW: an adaptive software architecture for the FFT

A new generation of the IMAGIC image processing system.

Although the formalism that allows the calculation of alloy thermodynamic properties from
first-principles has been known for decades, its practical implementation has so far remained a tedious
process. The Alloy Theoretic Automated Toolkit (ATAT) drastically simplifies this procedure by implementing
decision rules based on formal statistical analysis that frees the researchers from a constant
monitoring during the calculation process and automatically “glues” together the input and the output of
various codes, in order to provide a high-level interface to the calculation of alloy thermodynamic properties
from first-principles. ATAT implements the Structure Inversion Method (SIM), also known as the
Connolly-Williams method, in combination with semi-grand-canonical Monte Carlo simulations. In order
to make this powerful toolkit available to the wide community of researchers who could benefit from it,
this article present a concise user guide outlining the steps required to obtain thermodynamic information
from ab initio calculations.

/pdf/the-alloy-theoretic-automated-toolkit-a-user-guide-3hlxchpv7x.pdf

The Alloy Theoretic Automated Toolkit: A User Guide

Measurements of ground motion generated by nuclear explosions in Nevada were made for 37 locations near San Francisco Bay, California. The results were compared with the San Francisco 1906 earthquake intensities and the strong-motion recordings of the San Francisco earthquake of March 22, 1957. The recordings show marked amplitude variations which are related consistently to the geologic setting of the recording site. For sites underlain by a layer of younger bay mud or artificial fill, maximum horizontal ground velocities generally increased with thickness of the layer and were as much as ten times greater than those recorded on nearby bedrock. The maximum vertical velocities for these sites were between 1 and 3.5 times greater. Spectral amplification curves clearly define a “dominant ground period” of about 1 second for sites underlain by younger bay mud. For sites underlain by older, more consolidated sediments, no clearly defined “dominant ground period” was found. Maximum ground velocities for the older bay sediment sites were about twice those recorded on bedrock. Consistent correlations of the results from the nuclear recordings with the 1906 earthquake intensities and the spectral amplification curves for the 1957 earthquake suggest that areas of high amplification determined from small ground motions may also be areas of high intensity in future earthquakes.

Effects of local geology on ground motion near San Francisco Bay

This paper presents an algorithm for computing the fast Fourier transform, based on a method proposed by Cooley and Tukey. As in their algorithm, the dimension n of the transform is factored (if possible), and n/p elementary transforms of dimension p are computed for each factor p of n . An improved method of computing a transform step corresponding to an odd factor of n is given; with this method, the number of complex multiplications for an elementary transform of dimension p is reduced from (p-1)^{2} to (p-1)^{2}/4 for odd p . The fast Fourier transform, when computed in place, requires a final permutation step to arrange the results in normal order. This algorithm includes an efficient method for permuting the results in place. The algorithm is described mathematically and illustrated by a FORTRAN subroutine.

An algorithm for computing the mixed radix fast Fourier transform

A method is given for computing the fast Fourier transform of arbitrarily large size using auxiliary memory files, such as magnetic tape or disk, for data storage. Four data files are used, two in and two out. A multivariate complex Fourier transform of n=2^{m} data points is computed in m passes of the data, and the transformed result is permuted to normal order by m - 1 additional passes. With buffered input-output, computing can be overlapped with reading and writing of data. Computing time is proportional to n \log_{2} n . The method can be used with as few as three files, but file passing for permutation is reduced by using six or eight files. With eight files, the optimum number for a radix 2 transform, the transform is computed in m passes without need for additional permutation passes. An ALGOL procedure for computing the complex Fourier transform with four, six, or eight files is listed, and timing and accuracy test results are given. This procedure allows an arbitrary number of variables, each dimension a power of 2.

A method for computing the fast Fourier transform with auxiliary memory and limited high-speed storage

Poulter previously presented sonagram analyses of underwater sound recordings of various sea mammals, calling attention to the apparent harmonic progression of components of these signals. Other workers have questioned the presence of these harmonics in the original data and have suggested that they may instead be introduced during the analysis. In this paper we use underwater sound recordings of the male killer whale Namu, taken with equipment flat within ±2 dB to beyond 18 342 Hz, the upper limit of the analysis, to show by means of digital spectral analysis methods that a harmonic progression exists. In the signals analyzed, peaks in the estimated spectrum were observed at each integer multiple of the fundamental frequency. In view of the broad frequency response of the recording equipment and the precision of the subsequent digital analysis, we can say with confidence that these harmonics were actually present in the whaler's call. For the analysis, portions of an audio recording were converted to digital form. In the conversion, a bandpass filter was used to attenuate power below 40 Hz and above 10 kHz. Digital analysis techniques similar to those proposed by Bingham, Godfrey, and Tukey were then used. For each time span of data to be analyzed, a windowed Fourier transform was first computed, using a fast Fourier transform program. The power spectrum was next computed, as the squared modulus of the windowed transform, and a correction was made for the attenuation of high frequencies during A-to-D conversion. The autocorrelation function was estimated by computing the inverse transform of the power spectrum. A moderate amount of digital smoothing was then applied to the spectrum to reduce irregularities due to noise. The resulting smoothed spectrum is used as an estimate of the power spectral density function.

Spectral analysis of the call of the male killer whale

166 L. E. Alsop and A. A. Nowroozi, “Fast Fourier analysis,” J. Geophys. Res., vol. 71, pp. 5482-5483, November 15, 1966. €3. Andrews, “A high-speed algorithm for the computer generation of Fourier  transforms,” IEEE Trans. Computers (Short Notes), vol. C-17, pp. 373.375, April 1968. J. S . Bailey, “A fast  Fourier  transform  without multiplications,” Proc. Symp. on Computer Processing in Communications, vol. 19, MKI Symposia Ser. New York: Polytechnic Press, 1969. V. Benignus, “Estimation of the coherence spectrum and its confidence interval using the fast Fourier transform,” this issue, pp. 145-150. G. D. Bergland, “The fast Fourier transform recursive equations  for  arbitrary  length  records,” Math. Computation, vol. 21, pp, 236-238, April 1967. -9 “A fast Fourier transform algorithm using base eight iterations,” Math. Computation, vol. 22, pp. 275-279, April 1968. -, “A fast Fourier transform algorithm for realvalued series,” Commun. A C M , vol. 11, pp. 703--710, October 1968. -, “A radix-eight fast Fourier transform subroutine for real-valued series,” this issue, pp. 138144. -, “A guided tour of the fast Fourier transform,” IEEE Spectrum (to be published). “Fast  Fourier  transform  hardware implementations. I. An overview. 11. A survey,’’ this issue,

R. Singleton

Papers

An algorithm for computing the mixed radix fast Fourier transform

A method for computing the fast Fourier transform with auxiliary memory and limited high-speed storage

Spectral analysis of the call of the male killer whale

A short bibliography on the fast Fourier transform