James W. Cooley

The Fast Fourier Transform and Its Applications

The advent of the fast Fourier transform method has greatly extended our ability to implement Fourier methods on digital computers. A description of the alogorithm and its programming is given here and followed by a theorem relating its operands, the finite sample sequences, to the continuous functions they often are intended to approximate. An analysis of the error due to discrete sampling over finite ranges is given in terms of aliasing. Procedures for computing Fourier integrals, convolutions and lagged products are outlined.

The fast Fourier transform algorithm: Programming considerations in the calculation of sine, cosine and Laplace transforms☆

Abstract In the organization of programming packages for computing Fourier and Laplace transforms, it is useful, both for conceptual understanding and for operational efficiency to consider the discrete complex Fourier transform as a kind of nucleus around which programming for special applications is performed. An advantage of these procedures is that the basic complex Fourier transform algorithm is systematic and can relatively easily be implemented in efficient subroutines, micro-programs and special hardware devices. Once this is done, programming for special properties of the data can efficiently be left to the user to implement on a general purpose computer. The problem of establishing the correspondence between the discrete transforms and the continuous functions with which one is usually dealing is described. The application of these results and the above-mentioned subroutines to the calculation and inversion of Laplace transforms is given with formulas and empirical results displaying the effect of optimal parameters on computational efficiency and accuracy.

Application of the fast Fourier transform to computation of Fourier integrals, Fourier series, and convolution integrals

The fast Fourier transform is a computational procedure for calculating the finite Fourier transform of a time series. In this paper, the properties of the finite Fourier transform are related to commonly used integral transforms including the Fourier transform and convolution integrals. The relationship between the finite Fourier transform and Fourier series is also discussed.

The finite Fourier transform

The finite Fourier transform of a finite sequence is defined and its elementary properties are developed. The convolution and term-by-term product operations are defined and their equivalent operations in transform space are given. A discussion of the transforms of stretched and sampled functions leads to a sampling theorem for finite sequences. Finally, these results are used to give a simple derivation of the fast Fourier transform algorithm.

Historical notes on the fast Fourier transform

The fast Fourier transform algorithm has a long and interesting history that has only recently been appreciated. In this paper, the contributions of many investigators are described and placed in historical perspective.

Two algorithms for fast approximate subspace tracking

New fast algorithms are presented for tracking singular values, singular vectors, and the dimension of a signal subspace through an overlapping sequence of data matrices. The basic algorithm is called fast approximate subspace tracking (FAST). The algorithm is derived for the special case in which the matrix is changed by deleting the oldest column, shifting the remaining columns to the left, and adding a new column on the right. A second algorithm (FAST2) is specified by modifying FAST to trade reduced accuracy for higher speed. The speed and accuracy are compared with the PL algorithm, the PAST and PASTd algorithms, and the FST algorithm. An extension to multicolumn updates for the FAST algorithm is also discussed.

Teminology in digital signal processing

The committee on Digital Signal Processing of the IEEE Group on Audio and Electroacoustics has undertaken the project of recommending terminology for use in papers and texts on digital signal processing. The reasons for this project are twofold. First, the meanings of many terms that are commonly used differ from one author to another. Second, there are many terms that one would like to have defined for which no standard term currently exists. It is the purpose of this paper to propose terminology which we feel is self-consistent, and which is in reasonably good agreement with current practices. An alphabetic index of terms is included at the end of the paper.

Fourier transform and convolution subroutines for the IBM 3090 Vector facility

A set of highly optimized subroutines for digital signal processing has been included in the Engineering and Scientific Subroutine Library (ESSL) for the IBM 3090 Vector Facility. These include FORTRAN-callable subroutines for Fourier transforms, convolution, and correlation. The subroutines are carefully designed and tuned for optimal vector and cache performance. Speedups of up to 9½ times over scalar performance on the 3090 have been obtained.

The application of the fast Fourier transform algorithm to the estimation of spectra and cross-spectra

Abstract Alternative methods for the estimation of spectra are described and compared. Some of these methods have only become practicable with the advent of the fast Fourier transform algorithm although all of them benefit from the computational speed and accuracy of the algorithm. Topics discussed in relation to these methods for estimating spectra are spectral windows and bias in the estimates, general questions of statistical variability, the use of regression methods to smooth the periodogram, and use of time sectioning of the data to either smooth or to investigate non-stationarities in the data. The use of these methods and the fast Fourier transform algorithm for estimating cross-spectra is discussed briefly.

Discrete Fourier transforms and their applications

This text is designed to be a practial handbook on the evaluation and application of one of the major techniques for discrete signal processing. Knowledge of the discrete Fourier transform (DFT) and the ability to construct alogorithms based on the techniques of fast Fourier analysis are essential prerequisites for communications and cybernetics engineers. These methods are also of inestimable value to applied scientists in many other fields. The treatment given here is aimed specifically at such experimentalists and practitioners, and includes only such mathematical development as is necessary to give a feel for the significance of the methods, and to promote proficiency in its use. An introductory discourse on the general theory of Fourier series and transforms is followed by a thorough review of the properties and means of computation of the DFT. The fast Fourier transform is presented as a particularly efficient algorithm for DFT evaluation, and is described in some detail. Some applications of DFTs are discussed, and the book is rounded off with an introduction to discrete Hilbert transforms. Examples are provided throughout the text, and a full bibliography provides the basis for further study of the mathematical theory and specific areas of application.