Showing papers on "Harmonic wavelet transform published in 1967"
IBM1
TL;DR: In this article, the use of the fast Fourier transform in power spectrum analysis is described, and the method involves sectioning the record and averaging modified periodograms of the sections.
Abstract: The use of the fast Fourier transform in power spectrum analysis is described. Principal advantages of this method are a reduction in the number of computations and in required core storage, and convenient application in nonstationarity tests. The method involves sectioning the record and averaging modified periodograms of the sections.
9,705 citations
TL;DR: The discrete Fourier transform of a time series is defined, some of its properties are discussed, the associated fast method for computing this transform is derived, and some of the computational aspects of the method are presented.
Abstract: The fast Fourier transform is a computational tool which facilitates signal analysis such as power spectrum analysis and filter simulation by means of digital computers. It is a method for efficiently computing the discrete Fourier transform of a series of data samples (referred to as a time series). In this paper, the discrete Fourier transform of a time series is defined, some of its properties are discussed, the associated fast method (fast Fourier transform) for computing this transform is derived, and some of the computational aspects of the method are presented. Examples are included to demonstrate the concepts involved.
471 citations
TL;DR: The Fast Fourier Transform (FFT) as discussed by the authors is a computer algorithm that computes the discrete Fourier transform much faster than other algorithms, and it can be used to compute an N = 210-point transform 100 times faster than using a direct approach.
Abstract: The fast Fourier transform (FFT), a computer algorithm that computes the discrete Fourier transform much faster than other algorithms, is explained. Examples and detailed procedures are provided to assist the reader in learning how to use the algorithm. The savings in computer time can be huge; for example, an N = 210-point transform can be computed with the FFT 100 times faster than with the use of a direct approach.
271 citations
01 Oct 1967
TL;DR: The discrete Fourier transform of a time series is defined, some of its properties are discussed, the associated fast method for computing this transform is derived, and some of the computational aspects of the method are presented.
Abstract: The fast Fourier transform is a computational tool which facilitates signal analysis such as power spectrum analysis and filter simulation by means of digital computers. It is a method for efficiently computing the discrete Fourier transform of a series of data samples (referred to as a time series). In this paper, the discrete Fourier transform of a time series is defined, some of its properties are discussed, the associated fast method (fast Fourier transform) for computing this transform is derived, and some of the computational aspects of the method are presented. Examples are included to demonstrate the concepts involved.
217 citations
IBM1
TL;DR: The fast Fourier transform algorithm has a long and interesting history that has only recently been appreciated as discussed by the authors, and the contributions of many investigators are described and placed in historical perspective in this paper.
Abstract: 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.
196 citations
TL;DR: 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.
Abstract: and have shown major time savings in using it to compute large transforms on a digital computer. With n a power of two, computing time for this algorithm is proportional to n log2 n, a major improvement over other methods with computing time proportional to n 2. In this paper, the fast Fourier transform algorithm is briefly reviewed and fast difference equation methods for accurately computing the needed trigonometric function values are given. The problem of computing a large Fourier transform on a system with virtual memory is considered, and a solution is proposed. This method has been used to compute complex Fourier transforms of size n = 2 z6 on a computer with 215 words of core storage; this exceeds by a factor of eight the maximum radix two transform size with fixed allocation of this amount of core storage. The method has also been used to compute large mixed radix transforms. A scaling plan for computing the fast Fourier transform with fixed-point arithmetic is also given.
142 citations
IBM1
TL;DR: The contributions of many investigators are described and placed in historical perspective in this paper on the fast Fourier transform algorithm.
Abstract: 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.
121 citations
28 citations
23 citations
Patent•
24 Nov 1967TL;DR: In this article, the Cooley-Tukey algorithm is implemented in analog form, which significantly reduces the time needed to compute either the Fourier transform or the inverse transform of a signal segment.
Abstract: The recursive equations of the Cooley-Tukey algorithm are implemented in analog form, thereby significantly decreasing the time needed to compute either the Fourier transform or the inverse Fourier transform of a signal segment, relative to the time needed for the same computation by a digital implementation of these equations.
14 citations
TL;DR: In this article, relations for transforms of images in optical interference systems with incoherent illumination were obtained for shifted objects, and they were reduced to relations used for holography for shifting objects.
TL;DR: The modification of the algorithm reported here eliminates the extra passes through the data required by the additional zero interferogram points, thus reducing the computer time for N in put points and M output points to being proportional to M log2 N.
Abstract: The interferograms obtained in Fourier transform spectroscopy should be sampled with the minimum frequency allowed by the sampling theorem. This combines the best signal-to-noise ratio with the most efficient use of interferometer operation time. As is well known, all the information contained in the interferogram can be obtained by the calculation of the spectrum at a number of output points approximately equal to the number of points taken on the interferogram. To obtain a clearer picture of the details of the spectrum it is often desirable, however, to obtain the spec trum at more closely spaced intervals. Because of the long com puting times necessary with the usual Fourier transform pro grams, it was most efficient to calculate the spectrum by the Four ier transform at only the minimum number of points. Interpola tion with an appropriate apparatus function could then be car ried out to obtain intermediate points. With the very significant increase in computing speed possible with the Fourier transform algorithm developed by Cooley and Tukey and extended by Forman, it is now more efficient, in most applications, to calcu late the spectrum at more than the minimum number of points. However, the algorithm, as presented in both Refs. 3 and 4, pre supposes equal numbers of input and output points. Forman has pointed out that smaller output point spacing may be obtained by extending the interferogram with zeros. The computing time for M output points is then proportional to M log2 M. The modification of the algorithm reported here eliminates the extra passes through the data required by the additional zero interferogram points, thus reducing the computer time for N in put points and M output points to being proportional to M log2 N. We assume N = 2 samples, spaced Ax apart from the original interferogram. To obtain M = 2 output points, spaced Δv = (MΔx) apart, the interferogram is extended with P = 2 zeros, m, n, and p all being integers. Using Forman's notation, the spectrum, F(kΔv) = F(k), is
01 Jan 1967