Showing papers on "Non-uniform discrete Fourier transform published in 1979"
TL;DR: In this article, it was shown that there is really not much difference between pure and applied mathematics and that the difference between the two domains can be bridged through communication between the pure and the applied mathematicians.
Abstract: Let me begin with my view of a bit of history. Before the Second World War mathematics in the United States was a servant of the needs of others and mathematicians taught service courses. Indeed, while A. Weil was teaching at an Eastern university it would be only a slight exaggeration to say that he was forbidden from presenting proofs in class and was called on the carpet by a dean for breaking this structure. In the years after the War, mathematics became a subject in its own right. Proofs became acceptable, as the creation of the "new math" proved to the world. Mathematicians were in demand, were men in their own right and no one's servants. However, this growth period had a very unfortunate side affect. While mathematics was becoming a subject in its own right, many of its practitioners wanted to rid themselves of their former servant image. They had felt denigrated by the service role; so they denigrated service mathematics. Unfortunately, they lumped together service mathematics and applied mathematics. And so during this growth period of mathematics, there sprang up a distinction between pure and applied mathematics. During these years, the applied mathematicians felt the pure mathematicians looked down on them, and so the communications between the pure and applied mathematicians virtually dried up. In this paper we willl show that there is really not much difference between pure and applied mathematics. Indeed, we will cite instances of pure and applied mathematicians doing the same or analogous mathematics, but because of the lack of communication neither knew of the others' work. With these broad generalities stated, let me try to explain how I came to the writing of this paper. This may perhaps serve as an example of how the gap between pure and applied mathematicians can be bridged. I became interested in the study of the finite Fourier transform because I needed to know the eigenvalues of the finite Fourier transform. This arose in the study of the multiplicity of the regular representation of a solvmanifold. This problem was solved and the solution can be found in [8, p. 95]. Tolimieri, and Tolimieri and I, took up this problem in [18] and [3] and related the eigenvalue problem of the finite Fourier transform to a certain algebra of theta functions as discussed in Chapter I of this paper. I felt that
161 citations
83 citations
TL;DR: From this solution an efficient numerical procedure is derived for synthesis of discrete-interval, binary, periodic signals the Fourier amplitude spectrum of which is optimal in the same sense.
Abstract: A closed-form solution is presented for the discrete-interval, binary, periodic signal the complex Fourier coefficient spectrum of which optimally approximates in the least squares sense a desired complex Fourier coefficient spectrum. From this solution an efficient numerical procedure is derived for synthesis of discrete-interval, binary, periodic signals the Fourier amplitude spectrum of which is optimal in the same sense. Numerical examples show the practical feasibility of the procedure.
67 citations
IBM1
TL;DR: Some of the known results are extended to the case that some of the variables do satisfy some algebraic relations and applied to obtaining a lower bound on the multiplicative complexity of the Discrete Fourier Transform.
66 citations
01 Jan 1979
TL;DR: This paper presents two methods for computing discrete Fourier transforms (DFT) by polynomial transforms and shows that these techniques are particularly well adapted to multidimensional DFTs as well as to some one-dimensional DFT's and yield algorithms that are, in many instances, more efficient than the fast Fourier transform (FFT) or the Winograd Fourier Transform (WFTA).
64 citations
IBM1
TL;DR: In this article, two polynomial transforms have been proposed for computing discrete Fourier transform (DFT) by polynomials, which are particularly well adapted to multidimensional DFT's as well as to some one-dimensional DFTs.
Abstract: Polynomial transforms, defined in rings of polynomials, have been introduced recently and have been shown to give efficient algorithms for the computation of two-dimensional convolutions. In this paper we present two methods for computing discrete Fourier transforms (DFT) by polynomial transforms. We show that these techniques are particularly well adapted to multidimensional DFT's as well as to some one-dimensional DFT's and yield algorithms that are, in many instances, more efficient than the fast Fourier transform (FFT) or the Winograd Fourier Transform (WFTA). We also describe new split nesting and split prime factor techniques for computing large DFT's from a small set of short DFT's with a minimum number of operations.
63 citations
01 Jan 1979
TL;DR: In this paper, the authors describe the development of the numerical Fourier transform with exponential sampling (e.f.t.), which gives quite a high accuracy and a very long observation time with a small number of samples.
Abstract: The paper describes the development of the numerical Fourier transform with exponential sampling. This is called the exponential Fourier transform (e.f.t.). The e.f.t. gives quite a high accuracy and a very long observation time with a small number of samples in comparison with a conventional numerical Fourier transform with equally spaced sampling. Thus, the computational efficiency of the e.f.t. is much higher than the conventional Fourier transform. Various calculated examples using the e.f.t. are given and are compared with the results obtained by the conventional Fourier transform. From the results the superiority of the e.f.t. compared with the conventional Fourier transform becomes clear. The e.f.t. would be very useful to deal with electrical transients with a very wide range of frequency or time.
62 citations
TL;DR: The extremely simple mathematical technique called ''straightening through smoothing,'' which is a numerical frequency filter, is generalized in order to provide a transmission function having any shape.
Abstract: The extremely simple mathematical technique called ’’straightening through smoothing,’’ which is a numerical frequency filter, is generalized in order to provide a transmission function having any shape. This frequency filter requires such a small memory that it can be performed using a minicomputer or even a programmable hand held calculator and the number of channels used is not limited to a power of 2, as in the case of the fast Fourier transform. For some filtering functions the number of operations required is smaller than with the fast Fourier transform.
55 citations
TL;DR: A graph-theoretic model for a class of linear algorithms computing the discrete Fourier transform of sequences of length a power of 2 is presented and shown to be umquely optimal in tim class with respect to a naturally defined cost.
Abstract: A graph-theoretic model for a class of linear algorithms computing the discrete Fourier transform of sequences of length a power of 2, the mformat~on flow network, is presented The information flow network correspondmg to the fast Fourier transform IS shown to be umquely optimal in tim class with respect to a naturally defined cost
42 citations
TL;DR: It is shown that the discrete Walsh–Hadamardtransform applied to 2none-dimensional data is equivalent to the discrete n-dimensional Fourier transform applied to the same 2ndata arranged on the binary n-cube, which explains the theorem concerning the shift invariance of the power spectrum for the Walsh– hadamard transform and its generalizations.
Abstract: It is shown that the discrete Walsh–Hadamard transform applied to 2none-dimensional data is equivalent to the discrete n-dimensional Fourier transform applied to the same 2ndata arranged on the binary n-cube. A similar relationship is valid for the generalized discrete Walsh transform suggested by Andrews and Caspari. This relationship explains the theorem concerning the shift invariance of the power spectrum for the Walsh–Hadamard transform and its generalizations.
33 citations
TL;DR: Phase contrast photographs of diatoms are characterized from their Fourier transform taken through an optical diffractometer to find common features in a given set of di atoms.
Abstract: Phase contrast photographs of diatoms are characterized from their Fourier transform taken through an optical diffractometer. The system output is placed on line to a PDP11/40 providing digital subtraction of two output spectral distributions due to different species. Differences obtained in this manner are used for characterizing various species. An average Fourier transform taken through coherent additions is also analyzed to find common features in a given set of diatoms.
TL;DR: This paper shows how a similar approach can be used for sequences which are known to have only odd harmonics, and is shown to be essentially the dual of the known method for time symmetry.
Abstract: It is well known that if a finite duration, N-point sequence x(n) possesses certain symmetries, the computation of its discrete Fourier transform (DFT) can be obtained from an FFT of size N/2 or smaller. This is accomplished by first preprocessing the sequence, taking the FFT of the processed sequence, and then postprocessing the results to give the desired transform. In this paper we show how a similar approach can be used for sequences which are known to have only odd harmonics. The approach is shown to be essentially the dual of the known method for time symmetry. Computer programs are included for implementing the special procedures discussed in this paper.
TL;DR: Fourier transform of two-center charge distributions corresponding to arbitrary Slater-type orbitals are evaluated by a Gaussian quadrature procedure without any preliminary series expansion of the integrand as discussed by the authors.
Abstract: Fourier transform of two-center charge distributions corresponding to arbitrary Slater-type orbitals are evaluated by a Gaussian quadrature procedure without any preliminary series expansion of the integrand. Convergence and accuracy of the method are discussed and illustrated.
TL;DR: A modified version of the Winograd-Fourier transform algorithm is presented for use in transforming real vectors, using real arithmetic and real storage of intermediate results throughout while retaining the economy of Winog rad's basic method.
Abstract: A modified version of the Winograd-Fourier transform algorithm is presented for use in transforming real vectors. The new algorithm uses real arithmetic and real storage of intermediate results throughout while retaining the economy of Winograd's basic method. The derivation of the transform is explained and some programming techniques are discussed and illustrated.
01 Apr 1979
TL;DR: A more natural rate-pitch modification system using the constant-Q transform is presented which performs well with rate/pitch changes by factors of between one-third and three.
Abstract: Modification of the rate of occurrence of acoustic events without altering frequency content, and modification of pitch without changing time scale are presented as equivalent problems. While the short-time Fourier transform has been used to solve the rate modification problem, it is not a natural tool. It lacks the scaling property of the Fourier transform. The constant-Q transform, on the other hand, exhibits this property. A more natural rate/pitch modification system using the constant-Q transform is presented which performs well with rate/pitch changes by factors of between one-third and three.
TL;DR: The purpose of this correspondence is to point out a number of significant references, in the area of Walsh-Fourier transform conversion, that were missed in a recent paper.
Abstract: The purpose of this correspondence is to point out a number of significant references, in the area of Walsh-Fourier transform conversion, that were missed in a recent paper [1].
TL;DR: In this paper, a new s.a.w. time-inversion system is described, which requires fewer chirp-filter elements than those required with the common method of cascading two Fourier or Fresnel transformations.
Abstract: A new s.a.w. time-inversion system is described. It requires fewer chirp-filter elements than those required with the common method of cascading two Fourier or Fresnel transformations. This new method is based on the fact that the Fourier transform of a linear f.m. signal whose envelope is modulated by a given time function has (approximately) the time-inverted function as its amplitude. Theoretical discussion and some experimental results are included.
TL;DR: Transform methods for the interpolation of regularly spaced data are described, based on fast evaluation using discrete Fourier transforms, which produce an interpolation passing directly through the given values and are applied easily to the multi-dimensional case.
Abstract: Transform methods for the interpolation of regularly spaced data are described, based on fast evaluation using discrete Fourier transforms. For periodic data adequately sampled, the fast Fourier transform (FFT) is used directly. With undersampled or aperiodic data, a Chebyshev interpolating polynomial is evaluated by means of the FFT to provide minimum deviation and distributed ripple. The merits of two kinds of Chebyshev series are compared. All the methods described produce an interpolation passing directly through the given values and are applied easily to the multi-dimensional case.
IBM1
TL;DR: A new method is introduced for the fast computation of multidimensional discrete Fourier transforms that reduces significantly the number of operations over the conventional fast Fourier transform (f.f.t.) and is therefore attractive for image-processing applications.
Abstract: A new method is introduced for the fast computation of multidimensional discrete Fourier transforms (d.f.t.). We show that some multidimensional d.f.t.s are mapped efficiently into one-dimensional d.f.t.s by using a single polynomial transform and some auxiliary calculations. Since polynomial transforms can be computed without multiplications, this approach reduces significantly the number of operations over the conventional fast Fourier transform (f.f.t.) and is therefore attractive for image-processing applications.
TL;DR: In this article, a constant-Q digital spectral analysis scheme is described which exploits the perfect fifth symmetry of the 12-tone musical scale, which is a property exploited in our analysis.
Abstract: A constant‐Q digital spectral analysis scheme is described which exploits the ’’perfect fifth’’ symmetry of the 12‐tone musical scale.
TL;DR: The Fast Fourier Transform (FFT) for a step-like bounded function with unequal values at boundaries may be computed by using a convenient decomposition of the total curve into two elementary ones, one of them being a linear ramp as discussed by the authors.
Abstract: The calculation of the fast fourier transform (FFT) for a step-like bounded function with unequal values at boundaries may be performed by using a convenient decomposition of the total curve into two elementary ones, one of them being a linear ramp. The method may be generalized to functions having asymptotic tails which may be approximated by simple analytic functions, the theoretical FFT of which is known.
01 Jan 1979
TL;DR: In this article, the first n+1 values of a uniformly sampled Fourier transform of a real positive function are given, and it is shown that the Fourier terms of higher order can be used to estimate the brightness distribution.
Abstract: In radio astronomy it is often necessary to estimate a brightness distribution from a limited number of samples of its Fourier transform The manifest requirement that the brightness distribution be everywhere positive imposes definite constraints on its Fourier transform which yield information about unmeasured Fourier components Here we discuss the question: given the first n+1 values, P0, P1 … Pn, of a uniformly sampled Fourier transform of a real positive function, what can we say about Fourier terms of higher order?
IBM1
TL;DR: This paper presents two methods for computing discrete Fourier transforms (DFT) by polynomial transforms that are particularly well adapted to multidimensional DFTs and yield algorithms that are, in many instances, more efficient than the fast Fourier transform (FFT) or the Winograd Fourier Transform (WFTA).
Abstract: Polynomial transforms defined in rings of polynomials, have been introduced recently and shown to give efficient algorithms for the computation of two-dimensional convolutions. In this paper, we present two methods for computing discrete Fourier transforms (DFT) by polynomial transforms. We show that these techniques are particularly well adapted to multidimensional DFTs and yield algorithms that are, in many instances, more efficient than the fast Fourier transform (FFT) or the Winograd Fourier Transform (WFTA).
TL;DR: A multiddimensional fast-Fourier-transform algorithm is developed for the computation of multidimensional Fourier and Fourier-like discrete transforms; it has considerably less multiplications than the conventional fast-fourier -transform methods.
Abstract: A multidimensional fast-Fourier-transform algorithm is developed for the computation of multidimensional Fourier and Fourier-like discrete transforms; it has considerably less multiplications than the conventional fast-Fourier-transform methods.
01 Jan 1979
TL;DR: In this article, the use of MOS transistors and weighted capacitors in a device to calculate the transform of a set of signal samples in 300ns was discussed, and experimental results for a 16-point Hadamard and an 8-point complex Fourier transform were given.
Abstract: The use of MOS transistors and weighted capacitors in a device to calculate the transform of a set of signal samples in 300ns will be discussed. Experimental results for a 16-point Hadamard and an 8-point complex Fourier transform will be given.
TL;DR: Results are given of the sine and cosine transforms of a small circular aperture and an analysis shows this is due to unwanted circularly symmetrical moire patterns between the zone plates.
Abstract: A number of authors have pointed out that a system of zone plates combined with a diffuse source, transparent input, lens, and focusing screen will display on the output screen the Fourier transform of the input. Strictly speaking, the transform normally displayed is the cosine transform, and the bipolar output is superimposed on a dc gray level to give a positive-only intensity variation. By phase-shifting one zone plate the sine transform is obtained. Temporal modulation is possible. It is also possible to redesign the system to accept a diffusely reflecting input at the cost of introducing a phase gradient in the output. Results are given of the sine and cosine transforms of a small circular aperture. As expected, the sine transform is a uniform gray. Both transforms show unwanted artifacts beyond 0.1 rad off-axis. An analysis shows this is due to unwanted circularly symmetrical moire patterns between the zone plates.
TL;DR: It is pointed out that the method presented in the paper [1] is different from the methods referred to in the above paper, and the method has a small number of computations and inherently has no error.
Abstract: It is pointed out that the method presented in our paper [1] is different from the methods referred to in the above paper, and our method has a small number of computations and inherently has no error.
01 Oct 1979
TL;DR: In this paper, a discrete Bessel transform matrix which is asymptotically orthogonal has been proposed, whose orthogonality does not depend on its dimension.
Abstract: A discrete Bessel transform matrix which is asymptotically orthogonal has recently been proposed by Jerri [1]. In this letter we propose a new matrix whose orthogonality does not depend on its dimension.