[PDF] A short bibliography on the fast Fourier transform | Semantic Scholar (2024)

The author's own involvement and experience with the FFT algorithm is described, which led to an unfolding of its pre-electronic computer history going back to Gauss.

This chapter develops a fast algorithm, called the fast Fourier transform (FFT), which significantly reduces the number of arithmetic operations and memory required to compute the DFT (or its inverse).

It is shown how the Kronecker product can be mathematically defined and efficiently implemented using a matrix factorization method and a generalized spectral analysis is suggested, and a variety of examples are presented displaying various properties of the decompositions possible.

It is shown that a fast Fourier processor having an attractive cost performance ratio can be built by employing serial arithmetic in the implementation of the algorithm developed.

The design of a class of special-purpose computers for time-series analysis by Fourier transformation is described, which implement machine-oriented fast Fourier transform algorithms obtained by factoring the discrete Fouriertransform to an arbitrary radix.

Many transforms such as discrete Fourier, Hadamard, Haar etc and their innumerable generalizations have found wide applications in pattern recognition, signal processing, spectral analysis, etc and it is shown that all these transforms are particular cases of butterfly transforms, which use numerical or logical operations.

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The fast Fourier transform is a significant advance over previous methods, in that the number of arithmetic operations is proportional to n log2 n instead of n(2) and one that makes practical the solution of large problems in which data overlay within high speed storage will occur.

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.

This article describes another algorithm for computing the Discrete Fourier Transform where the required number of additions and subtractions is the same as in the Cooley-Tukey Algorithm; but therequired number of multiplications is only one half of that in the cooley- Tukey Al algorithm.

A theorem is proved that two methods for using the fast Fourier transform to reduce the number of arithmetic operations and, therefore the time required for computing discrete, preformulated, and finite convolutions can be modified to compute such difference equations.

The fast Fourier transform algorithm is briefly reviewed and fast difference equation methods for accurately computing the needed trigonometric function values are given and the problem of computing a large Fouriertransform on a system with virtual memory is considered, and a solution is proposed.

A modified version of the Fast Fourier Transform is developed and described and it is suggested that this form is of general use in the development and classification of various modifications and extensions of the algorithm.

A small-computer program has been written in this laboratory which uses the Danielson-Lanezos method with one minor modification, described below, and yields the same results as the binary form of the Cooley-Tukey algorithm with a comparable number of arithmetical operations.

Two digital machine organizations are suggested which use the fast Fourier transform algorithm for the cases of N (the number of samples analyzed) being a power of 2 and the case of N being the product of two integers.

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