Showing papers on "Fractional Fourier transform published in 1968"
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
TL;DR: 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.
Abstract: A modified version of the Fast Fourier Transform is developed and described. This version is well adapted for use in a special-purpose computer designed for the purpose. It is shown that only three operators are needed. One operator replaces successive pairs of data points by their sums and differences. The second operator performs a fixed permutation which is an ideal shuffle of the data. The third operator permits the multiplication of a selected subset of the data by a common complex multiplier.If, as seems reasonable, the slowest operation is the complex multiplications required, then, for reasonably sized date sets—e.g. 512 complex numbers—parallelization by the method developed should allow an increase of speed over the serial use of the Fast Fourier Transform by about two orders of magnitude.It is suggested that a machine to realize the speed improvement indicated is quite feasible.The analysis is based on the use of the Kronecker product of matrices. It is suggested that this form is of general use in the development and classification of various modifications and extensions of the algorithm.
362 citations
TL;DR: In this article, several basic power-spectrum estimation procedures are reviewed and their statistical and mathematical properties are discussed and compared with the standard procedure that uses the cosine transform of the estimated correlation function.
Abstract: The computation of power spectra, cross spectra, coherence, and bispectra of various types of geophysical random processes is part of the established routine. Since it is routine, some of the standard procedures need to be examined rather carefully to be certain that the assumptions behind the procedures are applicable to the data on hand. The basic criteria for a particular method are its resolution bandwidth, its variance, and its bias. In this paper several basic power-spectrum estimation procedures are reviewed and their statistical and mathematical properties are discussed. The direct use of the discrete Fourier transform for various spectrum calculations is discussed in detail, and its properties are compared with the standard procedure that uses the cosine transform of the estimated correlation function.
166 citations
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
09 Dec 1968
TL;DR: 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.
Abstract: With the advent of digital computers it became possible to compute the Discrete Fourier Transform for a large number of input points in relatively reasonable times. However, for certain uses a demand developed to compute the Discrete Fourier Transform in a very short time or even in real time. Also, a demand developed for computing the Fourier Transform for a very large number of input points. These demands resulted in a requirement for computing the Fourier Transform in the fastest time possible. A very economical way for computing the Fourier Transform was developed a few years ago and is known as the Cooley-Tukey Algorithm. 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 the required number of multiplications is only one half of that in the Cooley-Tukey Algorithm.
109 citations
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
01 Jan 1968
TL;DR: Several properties of the FHT are revealed, including the nature of its presence in the fast Fourier transform, in which it performs the additive operations as shown by further decomposing the product of matrices representing the FFT.
Abstract: : A discrete time transform was studied and applied to the representation and discrimination of digitized signals. The transform consists of an orthogonal (Hadamard) matrix whose elements are all ones and minus ones. To facilitate implementation, a fast Hadamard transform (FHT) has been developed requiring only NlogN rather than N squared algebraic additions. Several properties of the FHT are revealed, including the nature of its presence in the fast Fourier transform, in which it performs the additive operations as shown by further decomposing the product of matrices representing the FFT.
66 citations
TL;DR: The following procedures are based on the Cooley-Tukey algorithm for computing the finite Fourier transform of a complex data vector; the dimension of the data vector is assumed here to be a power of two.
Abstract: The following procedures are based on the Cooley-Tukey algorithm [1] for computing the finite Fourier transform of a complex data vector; the dimension of the data vector is assumed here to be a power of two. Procedure COMPLEXTRANSFORM computes either the complex Fourier transform or its inverse. Procedure REALTRANSFORM computes either the Fourier coefficients of a sequence of real data points or evaluates a Fourier series with given cosine and sine coefficients. The number of arithmetic operations for either procedure is proportional to n log2n, where n is the number of data points.
32 citations
TL;DR: A digital processor capable of computing the discrete Fourier transform for a range of audio signals in real time has been built as part of a facility to conduct research in signal processing.
Abstract: —A digital processor capable of computing the discrete Fourier transform for a range of audio signals in real time has been built as part of a facility to conduct research in signal processing. The digitized sample values can be complex. The arithmetic unit is configured to perform complex connectives, and automatic array scaling is used to make numerical accuracy independent of signal level. The Cooley–Tukey "fast Fourier transform" is the algorithm used.
25 citations
TL;DR: The following procedures are based on the Cooley-Tukey algor i thm for comput ing the finite Fourier t r ans fo rm of a complex da ta vector; the dimension of the da t a vector is assumed here to be a power of two.
Abstract: The following procedures are based on the Cooley-Tukey algor i thm [1] for comput ing the finite Fourier t r ans fo rm of a complex da ta vector; the dimension of the da t a vector is assumed here to be a power of two. Procedure COMPLEXTRANSFORM computes ei ther the complex Fourier t ransform or its inverse. Procedure REALTRANSFORM computes ei ther the Fourier coefficients of a sequence of real da ta points or eva lua tes a Fourier series wi th given cosine and sine coefficients. The number of ar i thmet ic operat ions for ei ther procedure is proport ional to n logs n, where n is the number of da ta points. Procedures FFT2, REVFFT2, REORDER, and REAL TRAN are building blocks, and are used in the two complete procedures ment ioned above. The fas t t r ans fo rm can be computed in a number of different ways, and these bui lding block procedures were wri t ten so as to make practical the comput ing of large t rans forms on a system wi th vir tual memory. Using a method proposed by Singleton [2], d a t a is accessed in sub-sequences of consecutive a r ray elements , and as m u ch comput ing as possible is done in one section of the d a t a before moving on to another. Procedure FFT2 computes the Fourier t r ans fo rm of da ta in normal order, giving a resu l t in reverse b ina ry order. Procedure REVFFT2 computes the Fourier t r ans fo rm of da ta in reverse b ina ry order and leaves the resul t in normal b inary order. Procedure REORDER permutes a complex vector f rom b inary to reverse b ina ry order or f rom reverse b inary to b inary order; this procedure also permutes real da ta in prepara t ion for efficient use of the complex Fourier t ransform. Procedures FFT2, REVFFT2, and REORDER m a y also be used to compute mul t iva r i a te Fourier t ransforms. The procedure R E A L T R A N is used to unscramble and combine the complex t rans forms of the even and odd numbered e lements of a sequence of real d a t a points . This procedure is not restr ic ted to powers of two and can be used whenever the number of da t a points is even.
23 citations
Patent•
28 Oct 1968
TL;DR: In this paper, a digital computer for rapidly determining the Fourier transform of a real input signal is disclosed, and the computer utilizes the symmetries of sinusoidal functions to reduce the computations required to determine a Fourier Transform.
Abstract: A digital computer for rapidly determining the Fourier transform of a real input signal is disclosed. The computer utilizes the symmetries of sinusoidal functions to reduce the computations required to determine the Fourier transform. Simultaneous addition, multiplication and memory accessing are performed by the computer thereby reducing the time normally required to compute a Fourier transform.
01 Oct 1968
TL;DR: An electrooptic light modulator (Pockels tube) and a coherent light source form the basis for the on-line generation of Fourier transforms of video images as discussed by the authors.
Abstract: An electrooptic light modulator (Pockels tube) and a coherent light source form the basis for the on-line generation of Fourier transforms of video images.
TL;DR: In this article, a technique is developed which greatly reduces the numerical labor involved in the Fourier Transform solution for cylindrical antennas of exbitrary length, which is then applied to solid antennas with two commonly used excitation geometries.
Abstract: A technique is developed in this paper which greatly reduces the numerical labor involved in the Fourier Transform solution for cylindrical antennas of exbitrary length. This method is then applied to solid antennas with two commonly used excitation geometries. It is also applied to the case of an antenna in a homogeneous conducting media. In all cases it is shown that the numerical results compare very well with the available experimental results.
TL;DR: In this article, the discrete Fourier transform is viewed as a set of discrete linear filters, one filter for each Fourier coefficient, and the characteristics of these filters are discussed.
Abstract: Computational procedures which have been developed in the past few years have taken the familiar frequency-domain techniques from the realm of theory and placed them in the realm of practice. In order to realize fully the potential of th techniques, it is necessary to gain insight into the physical significance of the discrete Fourier transform. Here, the discrete Fourier transform is viewed as a set of discrete linear filters--one filter for each Fourier coefficient. Each filter is seen to have zero poles and (N-1) zeros. (N is the number of data points transformed.) The characteristics of these filters are discussed. Spectrum weighting, for the purpose of sidelobe reduction, is also shown to be equivalent to discrete linear filtering. The filters in this case are similar to those which represent the discrete Fourier transform.
01 May 1968
TL;DR: In this article, it was shown that the Fourier transform of any continuous measure can be modified so that it becomes the transform of a singular measure in many groups, including the real line and the integers.
Abstract: not have a singular component. The present note contributes further evidence in this direction by showing that in many groups (including the real line and the integers) there are relatively small sets on which the Fourier transform of any absolutely continuous measure can be so modified that it becomes the transform of a singular measure. Let r be the dual of a locally compact abelian group G; L1(G) and M(G) denote the spaces of all Haar-integrable functions on G and of all complex Borel measures on G, respectively, and we identify L'(G) with the absolutely continuous members of M(G). The Fourier transform of ,u E M(G) is defined to be
CERN1
TL;DR: In this paper, the Fourier transform of a polynomially bounded function is expressed as a finite linear combination of analytic delta functionals applied to a continuous function on the real line in an improper sense.
Abstract: The Fourier transform of a product of two functions onSL(2,C) is expressed as a convolution integral of the Fourier transforms of its factors. With the help of this convolution integral we present the Fourier transform of a polynomially bounded function as a finite linear combination of analytic delta functionals applied to a continuous function on the real line in an improper sense.
TL;DR: For a simple evaluation of the Hilbert transform, iterated Fourier- or Laplace-transform schemes have been suggested as discussed by the authors, and an alternative iterated transform scheme, more general than the one previously reported, has been put forward.
Abstract: For a simple evaluation of the Hilbert transform, iterated Fourier- or Laplace-transform schemes have been suggested. In this letter, an alternative iterated transform scheme, more general than the one previously reported, has been put forward.
01 Jan 1968
09 Dec 1968
TL;DR: The development of the Fast Fourier Transform in complex notation has obscured the savings that can be made through the use of recursive properties of trigometric functions.
Abstract: The development of the Fast Fourier Transform in complex notation has obscured the savings that can be made through the use of recursive properties of trigometric functions. A disadvantage of the Fast Fourier Transform is that all samples of the function must be stored in memory before processing can start. The computation in the Fast Fourier Transform occurs after the receipt of the last sample of the function; there is no processing of the incoming data prior to this point. Thus if there are N samples of each function, and G different functions (in G "gates" or "channels"), then a total of GN words must be stored in memory.
TL;DR: In this article, character recognition by Incoherent Fourier Transformation (INFT) is presented. But the method is not suitable for character recognition in the real world, as shown in Figure 1.
Abstract: (1968). Character Recognition by Incoherent Fourier Transformation. Optica Acta: International Journal of Optics: Vol. 15, No. 6, pp. 627-628.
05 Jun 1968
TL;DR: A computer program is described which determines the Discrete Fourier Transform of a set of data, using a recently developed technique known as the Fast Fourier transform.
Abstract: : This research contribution describes a computer program (CNA Number 76-67) which determines the Discrete Fourier Transform of a set of data, using a recently developed technique known as the Fast Fourier transform. The relation between Discrete Fourier Transforms and Fourier Series when the data is periodic is also known.
TL;DR: In this article, a generalized finite integral transform combining the Fourier and Hankel transforms is introduced, together with a Laplace transformation with respect to time, making possible the simultaneous solution of the problems for a plate, a cylinder, and a sphere.
Abstract: A generalized finite integral transform combining the Fourier and Hankel transforms is introduced. This transform, together with a Laplace transformation with respect to time, makes possible the simultaneous solution of the problems for a plate, a cylinder, and a sphere.
TL;DR: In this article, a computationally oriented transform technique for the analysis of time-varying linear systems involving both discrete-time (i.e., sampled-data) variables and continuous-time variables is presented.
Abstract: A computationally oriented transform technique for the analysis of time-varying linear systems involving both discrete-time (i.e., sampled-data) variables and continuous-time variables is presented in this paper. Finite-dimensional representations are described for the various linear operators involved, and methods for obtaining singular-value decompositions of these operators are given. The three parts of the singular-value decomposition are analogous to the direct transform, transfer function, and inverse transform of Laplace and Fourier transforms, and offer analogous insights into analysis and synthesis problems. An example is included to illustrate the analysis procedures.